Sum-product problem over the reals

DefinitionSize of the larger of the sumset and the product set#✓fully proved

For a finite set \(A\) of real numbers, this is the natural number \(\max(|A+A|, |AA|)\), where \(A+A = \{a+b : a, b \in A\}\) is the sumset and \(AA = \{ab : a, b \in A\}\) is the product set of \(A\).

def sumProductMaxCard (A : Finset ℝ) : ℕ :=
  max (A + A).card (A * A).card

Used by 5

Used byUniform sum-product lower growth at rate \(s\) · The level minimum is attained · Sum-product ratio · The sum-product ratio is at most two · The sum-product ratio is nonnegative

DefinitionSum-product ratio#✓fully proved

For a finite set \(A\) of real numbers, the sum-product ratio of \(A\) is \(\log \max(|A+A|, |AA|) / \log |A|\), where \(A+A\) and \(AA\) are the sumset and the product set of \(A\). With the conventions \(\log 0 = 0\) and \(x / 0 = 0\), the ratio is \(0\) when \(|A| \le 1\).

def sumProductRatio (A : Finset ℝ) : ℝ :=
  Real.log (sumProductMaxCard A : ℝ) / Real.log (A.card : ℝ)

Uses 1 · Used by 6

Statement usesSize of the larger of the sumset and the product set

Used bySum-product ratios at a fixed cardinality · The level minimum is attained · The level minimum is at most two · The level minimum is nonnegative · The sum-product ratio is at most two · The sum-product ratio is nonnegative

DefinitionSum-product ratios at a fixed cardinality#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\). For a natural number \(n \ge 2\), this is the set of the real numbers \(\log \max(|A + A|, |AA|) / \log |A|\) for all sets \(A \subset \mathbb{R}\) with \(|A| = n\); for \(n \le 1\) it consists of the single number \(0\).

def fixedCardinalityRatios (n : ℕ) : Set ℝ :=
  {r : ℝ | ∃ A : Finset ℝ, A.card = n ∧ r = sumProductRatio A}

Uses 1 · Used by 4

Statement usesSum-product ratio

Used byLeast sum-product ratio at a fixed cardinality · The level minimum is attained · The level minimum is at most two · The level minimum is nonnegative

DefinitionLeast sum-product ratio at a fixed cardinality#✓fully proved

For a natural number \(n\), the level minimum \(m(n)\) is the infimum of the sum-product ratios \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, where \(A+A\) and \(AA\) are the sumset and the product set of \(A\), and the ratio is taken to be \(0\) when \(|A| \le 1\) (conventions \(\log 0 = 0\) and \(x / 0 = 0\)).

The set of these ratios is finite and nonempty, so the infimum is attained.

def sumProductLevelMinimum (n : ℕ) : ℝ :=
  sInf (fixedCardinalityRatios n)

Uses 1 · Used by 8

Statement usesSum-product ratios at a fixed cardinality

Used byThe sum-product level minima eventually lie in \((0, 2)\) · Sum-product exponent of the reals · The sum-product exponent ignores sets of size \(0\) and \(1\) · The level minima are eventually bounded above · The sum-product level minima are eventually bounded below · The level minimum is attained · The level minimum is at most two · The level minimum is nonnegative

TheoremThe level minimum is attained#✓fully proved

For every natural number \(n\), let \(m(n)\) be the infimum of the sum-product ratios \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, where \(A+A\) and \(AA\) are the sumset and the product set of \(A\) and the ratio is taken to be \(0\) when \(|A| \le 1\) (conventions \(\log 0 = 0\) and \(x / 0 = 0\)). Then \(m(n)\) is the least element of the set of these ratios: it is the ratio of some set of \(n\) real numbers, and no set of \(n\) real numbers has a smaller ratio.

theorem sumProductLevelMinimum_isLeast (n : ℕ) :
    IsLeast (fixedCardinalityRatios n) (sumProductLevelMinimum n)

Uses 4 · Used by 2

Statement usesSum-product ratios at a fixed cardinality · Least sum-product ratio at a fixed cardinality

Proof usesSize of the larger of the sumset and the product set · Sum-product ratio

Used byThe level minimum is at most two · The level minimum is nonnegative

TheoremThe sum-product ratio is nonnegative#✓fully proved

For every nonempty finite set \(A\) of real numbers, the sum-product ratio \(\log \max(|A+A|, |AA|) / \log |A|\) is at least \(0\), where \(A+A\) and \(AA\) are the sumset and the product set of \(A\), and the ratio is taken to be \(0\) when \(|A| = 1\) (conventions \(\log 0 = 0\) and \(x / 0 = 0\)).

theorem sumProductRatio_nonneg (A : Finset ℝ) (hA : A.Nonempty) :
    0 ≤ sumProductRatio A

Uses 2 · Used by 1

Statement usesSum-product ratio

Proof usesSize of the larger of the sumset and the product set

Used byThe level minimum is nonnegative

TheoremThe sum-product ratio is at most two#✓fully proved

For every finite set \(A\) of real numbers with \(|A| \ge 2\), the sum-product ratio \(\log \max(|A+A|, |AA|) / \log |A|\) is at most \(2\), where \(A+A\) and \(AA\) are the sumset and the product set of \(A\).

theorem sumProductRatio_le_two (A : Finset ℝ) (hA : 2 ≤ A.card) :
    sumProductRatio A ≤ 2

Uses 2 · Used by 1

Statement usesSum-product ratio

Proof usesSize of the larger of the sumset and the product set

Used byThe level minimum is at most two

TheoremThe level minimum is nonnegative#✓fully proved

For every natural number \(n \ge 1\), the level minimum \(m(n)\), the infimum of the sum-product ratios \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, satisfies \(m(n) \ge 0\). Here \(A+A\) and \(AA\) are the sumset and the product set of \(A\), and the ratio is taken to be \(0\) when \(|A| = 1\) (conventions \(\log 0 = 0\), \(x / 0 = 0\)).

theorem sumProductLevelMinimum_nonneg (n : ℕ) (hn : 0 < n) :
    0 ≤ sumProductLevelMinimum n

Uses 5 · Used by 1

Statement usesLeast sum-product ratio at a fixed cardinality

Proof usesSum-product ratios at a fixed cardinality · The level minimum is attained · Sum-product ratio · The sum-product ratio is nonnegative

Used byThe sum-product level minima eventually lie in \((0, 2)\)

TheoremThe level minimum is at most two#✓fully proved

For every natural number \(n \ge 2\), the level minimum \(m(n)\), the infimum of the sum-product ratios \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, where \(A+A\) and \(AA\) are the sumset and the product set of \(A\), satisfies \(m(n) \le 2\).

theorem sumProductLevelMinimum_le_two (n : ℕ) (hn : 2 ≤ n) :
    sumProductLevelMinimum n ≤ 2

Uses 5 · Used by 1

Statement usesLeast sum-product ratio at a fixed cardinality

Proof usesSum-product ratios at a fixed cardinality · The level minimum is attained · Sum-product ratio · The sum-product ratio is at most two

Used byThe sum-product level minima eventually lie in \((0, 2)\)

TheoremThe sum-product level minima eventually lie in \((0, 2)\)#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\). For a natural number \(n \ge 2\) let \(m_n\) be the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). Then \(0 \le m_n \le 2\) for every sufficiently large \(n\).

theorem eventually_sumProductLevelMinimum_mem_Icc :
    ∀ᶠ n : ℕ in Filter.atTop,
      sumProductLevelMinimum n ∈ Set.Icc (0 : ℝ) 2

Uses 3 · Used by 2

Statement usesLeast sum-product ratio at a fixed cardinality

Proof usesThe level minimum is at most two · The level minimum is nonnegative

Used byThe level minima are eventually bounded above · The sum-product level minima are eventually bounded below

TheoremThe level minima are eventually bounded above#✓fully proved

There is a real number \(b\) such that \(m(n) \le b\) for all sufficiently large natural numbers \(n\), where the level minimum \(m(n)\) is the infimum of the sum-product ratios \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, and \(A+A\) and \(AA\) are the sumset and the product set of \(A\).

theorem sumProductLevelMinimum_isBoundedUnder_le :
    Filter.atTop.IsBoundedUnder (· ≤ ·) sumProductLevelMinimum

Uses 2

Statement usesLeast sum-product ratio at a fixed cardinality

Proof usesThe sum-product level minima eventually lie in \((0, 2)\)

TheoremThe sum-product level minima are eventually bounded below#✓fully proved

There is a real number \(b\) such that the sum-product level minimum \(m(n)\) satisfies \(m(n) \ge b\) for all sufficiently large \(n\).

theorem sumProductLevelMinimum_isBoundedUnder_ge :
    Filter.atTop.IsBoundedUnder (· ≥ ·) sumProductLevelMinimum

Uses 2

Statement usesLeast sum-product ratio at a fixed cardinality

Proof usesThe sum-product level minima eventually lie in \((0, 2)\)

DefinitionSum-product exponent of the reals#✓fully proved

For \(n \in \mathbb{N}\) let \(m(n)\) be the sum-product level minimum: the least value of \(\log \max(|A + A|, |AA|) / \log |A|\) over sets \(A \subset \mathbb{R}\) with \(n\) elements, where \(A + A\) and \(AA\) are the sets of sums \(a + b\) and products \(ab\) of elements \(a, b \in A\). The real sum-product exponent is \(\liminf_{n \to \infty} m(n)\).

For \(n \le 1\) the conventions \(\log 0 = 0\) and \(x / 0 = 0\) give \(m(n) = 0\); these levels do not affect the limit inferior.

def sumProductExponent : ℝ :=
  Filter.liminf sumProductLevelMinimum Filter.atTop

Uses 1 · Used by 14

Statement usesLeast sum-product ratio at a fixed cardinality

Used byStatement of Bloom's lower bound for the real sum-product exponent · Statement of the Bloom–Sawin–Schildkraut–Zhelezov numerical upper bound · Statement of the Bloom–Sawin–Schildkraut–Zhelezov refutation · Statement of the Bloom–Sawin–Schildkraut–Zhelezov upper bound · Statement of Cushman's lower bound for the real sum-product exponent · Statement of Elekes's lower bound · Statement of the uniform-growth characterization of the real sum-product exponent · Statement of the Erdős–Szemerédi conjecture for real sets · Statement of the upper bound two for the real sum-product exponent · Statement of the Rudnev–Stevens lower bound · Statement of Solymosi's \(4/3\) lower bound · Statement of Solymosi's \(14/11\) lower bound · The sum-product exponent ignores sets of size \(0\) and \(1\) · Statement of the trivial lower bound for the real sum-product exponent

TheoremThe sum-product exponent ignores sets of size \(0\) and \(1\)#✓fully proved

Let \(m(n)\) be the sum-product level minimum at \(n\). The real sum-product exponent equals \(\liminf_{n \to \infty} m(n + 2)\), so the levels \(n = 0\) and \(n = 1\) do not affect it.

theorem sumProductExponent_eq_liminf_add_two :
    sumProductExponent =
      Filter.liminf (fun n : ℕ => sumProductLevelMinimum (n + 2)) Filter.atTop

Uses 2

Statement usesSum-product exponent of the reals · Least sum-product ratio at a fixed cardinality

DefinitionUniform sum-product lower growth at rate \(s\)#✓fully proved

A real number \(s\) has uniform sum-product lower growth when for every \(\varepsilon > 0\) there is a constant \(K > 0\) such that every nonempty finite set \(A \subset \mathbb{R}\) satisfies \(K \cdot |A|^{s - \varepsilon} \le \max(|A + A|, |AA|)\), where \(A + A\) and \(AA\) are the sets of sums \(a + b\) and products \(ab\) of elements \(a, b \in A\).

The constant \(K\) may depend on \(\varepsilon\) but not on \(A\). Requiring \(A\) to be nonempty avoids giving a meaning to \(0^{s - \varepsilon}\); singleton sets are included.

def HasUniformSumProductLowerGrowth (s : ℝ) : Prop :=
  ∀ ε : ℝ, 0 < ε →
    ∃ K : ℝ, 0 < K ∧
      ∀ A : Finset ℝ,
        A.Nonempty →
          K * Real.rpow (A.card : ℝ) (s - ε) ≤ (sumProductMaxCard A : ℝ)

Uses 1 · Used by 1

Statement usesSize of the larger of the sumset and the product set

Used byStatement of the uniform-growth characterization of the real sum-product exponent

DefinitionStatement of the uniform-growth characterization of the real sum-product exponent#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). This is the proposition that \(C_{84b}\) is the largest real number \(s\) with the following property: for every \(\varepsilon > 0\) there is a constant \(K > 0\) such that \(K |A|^{s - \varepsilon} \le \max(|A + A|, |AA|)\) for every nonempty finite set \(A \subset \mathbb{R}\).

def epsilonUniformLowerGrowthCharacterization : Prop :=
  IsGreatest {s : ℝ | HasUniformSumProductLowerGrowth s} sumProductExponent

Uses 2 · Used by 1

Statement usesUniform sum-product lower growth at rate \(s\) · Sum-product exponent of the reals

Used byThe real sum-product exponent is the largest uniform growth rate

DefinitionStatement of the Erdős–Szemerédi conjecture for real sets#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). This is the proposition that \(C_{84b} = 2\).

def erdosSzemerediHistoricalConjecture : Prop :=
  sumProductExponent = 2

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byThe Erdős–Szemerédi conjecture fails for real sets

DefinitionStatement of the Bloom–Sawin–Schildkraut–Zhelezov refutation#✓fully proved

The statement that the real sum-product exponent \(C\) satisfies \(C < 2\).

def bloomSawinSchildkrautZhelezovRefutation : Prop :=
  sumProductExponent < 2

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byThe real sum-product exponent is less than 2 (Bloom–Sawin–Schildkraut–Zhelezov)

DefinitionStatement of the upper bound two for the real sum-product exponent#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). This is the proposition that \(C_{84b} \le 2\).

def erdosSzemerediHistoricalUpperBound : Prop :=
  sumProductExponent ≤ 2

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byThe real sum-product exponent is at most two

DefinitionStatement of the Bloom–Sawin–Schildkraut–Zhelezov upper bound#✓fully proved

The statement that there is a real number \(c > 0\) such that the real sum-product exponent \(C\) satisfies \(C \le 2 - c\).

def bloomSawinSchildkrautZhelezovUpperBound : Prop :=
  ∃ c : ℝ, 0 < c ∧ sumProductExponent ≤ 2 - c

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byBloom–Sawin–Schildkraut–Zhelezov upper bound for the real sum-product exponent

DefinitionStatement of the Bloom–Sawin–Schildkraut–Zhelezov numerical upper bound#✓fully proved

The statement that the real sum-product exponent \(C\) satisfies \(C \le 199999913/100000000\).

def bloomSawinSchildkrautZhelezovNumericalUpperBound : Prop :=
  sumProductExponent ≤ (199999913 : ℝ) / 100000000

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byBloom–Sawin–Schildkraut–Zhelezov numerical upper bound for the real sum-product exponent

DefinitionStatement of the trivial lower bound for the real sum-product exponent#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\). This is the proposition that the real sum-product exponent, the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\), is at least \(1\).

def trivialLowerBound : Prop :=
  (1 : ℝ) ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byTrivial lower bound for the real sum-product exponent

DefinitionStatement of Elekes's lower bound#✓fully proved

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). This is the proposition that \(C_{84b} \ge 5/4\).

def elekesLowerBound : Prop :=
  (5 : ℝ) / 4 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byElekes's lower bound for the real sum-product exponent

DefinitionStatement of Solymosi's \(14/11\) lower bound#✓fully proved

The proposition that the real sum-product exponent is at least \(14/11\).

def solymosiFourteenEleventhsLowerBound : Prop :=
  (14 : ℝ) / 11 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used bySolymosi's \(14/11\) lower bound for the real sum-product exponent

DefinitionStatement of Solymosi's \(4/3\) lower bound#✓fully proved

The proposition that the real sum-product exponent is at least \(4/3\).

def solymosiFourThirdsLowerBound : Prop :=
  (4 : ℝ) / 3 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used bySolymosi's \(4/3\) lower bound for the real sum-product exponent

DefinitionStatement of the Rudnev–Stevens lower bound#✓fully proved

The proposition that the real sum-product exponent is at least \(1558/1167 = 4/3 + 2/1167\).

def rudnevStevensLowerBound : Prop :=
  (1558 : ℝ) / 1167 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byRudnev–Stevens lower bound for the real sum-product exponent

DefinitionStatement of Bloom's lower bound for the real sum-product exponent#✓fully proved

The statement that the real sum-product exponent \(C\) satisfies \(1270/951 \le C\).

def bloomLowerBound : Prop :=
  (1270 : ℝ) / 951 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byBloom's lower bound for the real sum-product exponent

DefinitionStatement of Cushman's lower bound for the real sum-product exponent#✓fully proved

The statement that the real sum-product exponent \(C\) satisfies \(4/3 + 10/4407 \le C\).

def cushmanCurrentLowerBound : Prop :=
  (4 : ℝ) / 3 + (10 : ℝ) / 4407 ≤ sumProductExponent

Uses 1 · Used by 1

Statement usesSum-product exponent of the reals

Used byCushman's lower bound for the real sum-product exponent

TheoremThe real sum-product exponent is the largest uniform growth rate#●ready to prove

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). Then \(C_{84b}\) is the largest real number \(s\) with the following property: for every \(\varepsilon > 0\) there is a constant \(K > 0\) such that \(K |A|^{s - \varepsilon} \le \max(|A + A|, |AA|)\) for every nonempty finite set \(A \subset \mathbb{R}\).

Optimization Problems 84b gives this as an equivalent definition of \(C_{84b}\).

public theorem_wanted epsilonUniformLowerGrowthCharacterization_holds : epsilonUniformLowerGrowthCharacterization

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the uniform-growth characterization of the real sum-product exponent

TheoremThe Erdős–Szemerédi conjecture fails for real sets#●ready to prove

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). Then \(C_{84b} \ne 2\).

Bloom, Sawin, Schildkraut and Zhelezov construct arbitrarily large finite sets \(A \subset \mathbb{R}\) with \(\max(|A + A|, |AA|) \le |A|^{2 - c}\) for an absolute constant \(c > 0\), so \(C_{84b} \le 2 - c\).

public theorem_wanted erdosSzemerediHistoricalConjecture_refuted : ¬ erdosSzemerediHistoricalConjecture

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the Erdős–Szemerédi conjecture for real sets

TheoremThe real sum-product exponent is less than 2 (Bloom–Sawin–Schildkraut–Zhelezov)#●ready to prove

The real sum-product exponent \(C\) satisfies \(C < 2\). Here, for a finite set \(A \subset \mathbb{R}\), \(A + A\) and \(A \cdot A\) are the sets of sums \(a + b\) and products \(ab\) with \(a, b \in A\), and \(M(A) = \max(|A + A|, |A \cdot A|)\); for \(n \ge 2\), \(m(n)\) is the minimum of \(\log M(A) / \log |A|\) over all \(A \subset \mathbb{R}\) with \(|A| = n\); and \(C = \liminf_{n \to \infty} m(n)\).

This refutes the historical Erdős–Szemerédi conjecture \(C = 2\) for real sets: Bloom, Sawin, Schildkraut and Zhelezov (arXiv:2605.28781) construct arbitrarily large finite sets \(A \subset \mathbb{R}\) with \(M(A) \le |A|^{2 - c}\) for an absolute constant \(c > 0\).

public theorem_wanted bloomSawinSchildkrautZhelezovRefutation_holds : bloomSawinSchildkrautZhelezovRefutation

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the Bloom–Sawin–Schildkraut–Zhelezov refutation

TheoremThe real sum-product exponent is at most two#●ready to prove

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). Then \(C_{84b} \le 2\).

This follows from \(|A + A| \le |A|^2\) and \(|AA| \le |A|^2\); Optimization Problems 84b records it as the upper bound of Erdős and Szemerédi (1983).

public theorem_wanted erdosSzemerediHistoricalUpperBound_holds : erdosSzemerediHistoricalUpperBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the upper bound two for the real sum-product exponent

TheoremBloom–Sawin–Schildkraut–Zhelezov upper bound for the real sum-product exponent#●ready to prove

There is a real number \(c > 0\) such that the real sum-product exponent \(C\) satisfies \(C \le 2 - c\). Here, for a finite set \(A \subset \mathbb{R}\), \(A + A\) and \(A \cdot A\) are the sets of sums \(a + b\) and products \(ab\) with \(a, b \in A\), and \(M(A) = \max(|A + A|, |A \cdot A|)\); for \(n \ge 2\), \(m(n)\) is the minimum of \(\log M(A) / \log |A|\) over all \(A \subset \mathbb{R}\) with \(|A| = n\); and \(C = \liminf_{n \to \infty} m(n)\).

Bloom, Sawin, Schildkraut and Zhelezov (arXiv:2605.28781) construct arbitrarily large finite sets \(A \subset \mathbb{R}\) with \(M(A) \le |A|^{2 - c}\) for an absolute constant \(c > 0\); Optimization Problems 84b lists the resulting upper bound \(2 - c\).

public theorem_wanted bloomSawinSchildkrautZhelezovUpperBound_holds : bloomSawinSchildkrautZhelezovUpperBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the Bloom–Sawin–Schildkraut–Zhelezov upper bound

TheoremBloom–Sawin–Schildkraut–Zhelezov numerical upper bound for the real sum-product exponent#●ready to prove

The real sum-product exponent \(C\) satisfies \(C \le 199999913/100000000 = 1.99999913\). Here, for a finite set \(A \subset \mathbb{R}\), \(A + A\) and \(A \cdot A\) are the sets of sums \(a + b\) and products \(ab\) with \(a, b \in A\), and \(M(A) = \max(|A + A|, |A \cdot A|)\); for \(n \ge 2\), \(m(n)\) is the minimum of \(\log M(A) / \log |A|\) over all \(A \subset \mathbb{R}\) with \(|A| = n\); and \(C = \liminf_{n \to \infty} m(n)\).

Bloom, Sawin, Schildkraut and Zhelezov sketch, in the section "Numerical estimates" of arXiv:2605.28781, a quantified version of their construction that gives arbitrarily large finite sets \(A \subset \mathbb{R}\) with \(M(A) \le |A|^{2 - 0.00000087}\); they call this value non-optimized. Optimization Problems 84b lists it as the explicit upper bound \(1.99999913\).

public theorem_wanted bloomSawinSchildkrautZhelezovNumericalUpperBound_holds : bloomSawinSchildkrautZhelezovNumericalUpperBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the Bloom–Sawin–Schildkraut–Zhelezov numerical upper bound

TheoremTrivial lower bound for the real sum-product exponent#●ready to prove

The real sum-product exponent is at least \(1\). Here the exponent is \(\liminf_{n \to \infty} m(n)\), where \(m(n)\) is the infimum of \(\log \max(|A+A|, |AA|) / \log |A|\) over all sets \(A\) of \(n\) real numbers, and \(A+A\) and \(AA\) are the sumset and the product set of \(A\).

Optimization Problems 84b lists this bound as trivial, since \(|A+A| \ge 2|A| - 1\).

public theorem_wanted trivialLowerBound_holds : trivialLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the trivial lower bound for the real sum-product exponent

TheoremElekes's lower bound for the real sum-product exponent#●ready to prove

For a finite set \(A \subset \mathbb{R}\) write \(A + A\) and \(AA\) for the sets of all sums \(a + b\) and all products \(ab\) with \(a, b \in A\), and let the real sum-product exponent \(C_{84b}\) be the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log n\) over sets \(A \subset \mathbb{R}\) with \(|A| = n\). Then \(C_{84b} \ge 5/4\).

Elekes proved, using the Szemerédi–Trotter theorem, that \(\max(|A + A|, |AA|) \gg |A|^{5/4}\) for finite sets \(A \subset \mathbb{R}\).

public theorem_wanted elekesLowerBound_holds : elekesLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of Elekes's lower bound

TheoremSolymosi's \(14/11\) lower bound for the real sum-product exponent#●ready to prove

Let \(C\) be the real sum-product exponent: the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log |A|\) over \(n\)-element sets \(A \subset \mathbb{R}\), where \(A + A\) and \(AA\) are the sets of sums \(a + b\) and products \(ab\) of elements \(a, b \in A\). Then \(C \ge 14/11\).

Optimization Problems 84b lists the lower bound \(14/11\) [So05] (Solymosi, Bull. London Math. Soc. 37 (2005)); Bloom's history lists Solymosi 2005 with exponent \(1 + 3/11\).

public theorem_wanted solymosiFourteenEleventhsLowerBound_holds : solymosiFourteenEleventhsLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of Solymosi's \(14/11\) lower bound

TheoremSolymosi's \(4/3\) lower bound for the real sum-product exponent#●ready to prove

Let \(C\) be the real sum-product exponent: the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log |A|\) over \(n\)-element sets \(A \subset \mathbb{R}\), where \(A + A\) and \(AA\) are the sets of sums \(a + b\) and products \(ab\) of elements \(a, b \in A\). Then \(C \ge 4/3\).

Optimization Problems 84b lists the lower bound \(4/3\) [So09] (Solymosi, Adv. Math. 222 (2009)); Bloom's history lists Solymosi 2009 with exponent \(1 + 1/3\).

public theorem_wanted solymosiFourThirdsLowerBound_holds : solymosiFourThirdsLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of Solymosi's \(4/3\) lower bound

TheoremRudnev–Stevens lower bound for the real sum-product exponent#●ready to prove

Let \(C\) be the real sum-product exponent: the limit inferior, as \(n \to \infty\), of the least value of \(\log \max(|A + A|, |AA|) / \log |A|\) over \(n\)-element sets \(A \subset \mathbb{R}\), where \(A + A\) and \(AA\) are the sets of sums \(a + b\) and products \(ab\) of elements \(a, b \in A\). Then \(C \ge 1558/1167\).

Rudnev and Stevens (arXiv:2005.11145, abstract) prove \(\max(|A + A|, |AA|) \ge |A|^{4/3 + 2/1167 - o(1)}\) for finite \(A \subset \mathbb{R}\), and \(4/3 + 2/1167 = 1558/1167\); Optimization Problems 84b lists this bound [RuSt22].

public theorem_wanted rudnevStevensLowerBound_holds : rudnevStevensLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of the Rudnev–Stevens lower bound

TheoremBloom's lower bound for the real sum-product exponent#●ready to prove

The real sum-product exponent \(C\) is at least \(1270/951\). Here, for a finite set \(A \subset \mathbb{R}\), \(A + A\) and \(A \cdot A\) are the sets of sums \(a + b\) and products \(ab\) with \(a, b \in A\), and \(M(A) = \max(|A + A|, |A \cdot A|)\); for \(n \ge 2\), \(m(n)\) is the minimum of \(\log M(A) / \log |A|\) over all \(A \subset \mathbb{R}\) with \(|A| = n\); and \(C = \liminf_{n \to \infty} m(n)\).

Optimization Problems 84b lists this lower bound as Bloom 2025; Bloom's history of the sum-product problem attributes it to Bloom, Control and its applications in additive combinatorics. It also follows from Cushman's larger lower bound \(4/3 + 10/4407\).

public theorem_wanted bloomLowerBound_holds : bloomLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of Bloom's lower bound for the real sum-product exponent

TheoremCushman's lower bound for the real sum-product exponent#●ready to prove

The real sum-product exponent \(C\) is at least \(4/3 + 10/4407\). Here, for a finite set \(A \subset \mathbb{R}\), \(A + A\) and \(A \cdot A\) are the sets of sums \(a + b\) and products \(ab\) with \(a, b \in A\), and \(M(A) = \max(|A + A|, |A \cdot A|)\); for \(n \ge 2\), \(m(n)\) is the minimum of \(\log M(A) / \log |A|\) over all \(A \subset \mathbb{R}\) with \(|A| = n\); and \(C = \liminf_{n \to \infty} m(n)\).

Cushman (arXiv:2512.13849) proves \(M(A) \gg_\varepsilon |A|^{4/3 + 10/4407 - \varepsilon}\) for every finite \(A \subset \mathbb{R}\); Optimization Problems 84b lists this as the current record.

public theorem_wanted cushmanCurrentLowerBound_holds : cushmanCurrentLowerBound

Sources

• Paul Erdős and Endre Szemerédi, On sums and products of integers, Studies in Pure Mathematics (1983), 213-218, https://doi.org/10.1007/978-3-0348-5438-2_19; György Elekes, On the number of sums and products, Acta Arithmetica 81 (1997), no. 4, 365-367, https://doi.org/10.4064/aa-81-4-365-367; József Solymosi, On the number of sums and products, Bulletin of the London Mathematical Society 37 (2005), no. 4, 491-494, https://doi.org/10.1112/S0024609305004261; József Solymosi, Bounding multiplicative energy by the sumset, Advances in Mathematics 222 (2009), no. 2, 402-408, https://doi.org/10.1016/j.aim.2009.04.006; Misha Rudnev and Sophie Stevens, An update on the sum-product problem, Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411-430, https://arxiv.org/abs/2005.11145; Thomas F. Bloom, A history of the sum-product problem, https://thomasbloom.org/notes/sumproduct.html; Adam Cushman, A note on the sum-product problem and the convex sumset problem, arXiv:2512.13849 (2025), https://arxiv.org/abs/2512.13849; Thomas F. Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026), https://arxiv.org/abs/2605.28781; Optimization Problems, record 84b, Sum-product exponent for the reals, tables of known upper and lower bounds, https://github.com/teorth/optimizationproblems/blob/2c1968cd520b60f1cf3cf50749f7285e800e4e15/constants/84b.md.

Uses 1

Statement usesStatement of Cushman's lower bound for the real sum-product exponent

28 fully proved13 ready to prove

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