Sums of dilates

DefinitionLower bound for sums of transcendental dilates#✓fully proved

There is a real constant \(c > 0\) such that for every real number \(\alpha\) transcendental over \(\mathbb{Q}\) and every finite set \(A\) of real numbers,

\[e^{c\sqrt{\log |A|}}\,|A| \le |A + \alpha \cdot A|,\]

where \(A + \alpha \cdot A = \{a + \alpha b : a, b \in A\}\).

def transcendentalDilateLowerBound : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∀ (α : ℝ), Transcendental ℚ α → ∀ A : Finset ℝ,
    Real.exp (c * Real.sqrt (Real.log A.card)) * A.card ≤ ((A + α • A).card : ℝ)

Used by 1

Used byConlon–Lim theorem on sums of transcendental dilates

TheoremConlon–Lim theorem on sums of transcendental dilates#●ready to prove

There is a real constant \(c > 0\) such that for every real number \(\alpha\) transcendental over \(\mathbb{Q}\) and every finite set \(A\) of real numbers,

\[e^{c\sqrt{\log |A|}}\,|A| \le |A + \alpha \cdot A|,\]

where \(A + \alpha \cdot A = \{a + \alpha b : a, b \in A\}\).

Resolved true: Conlon and Lim prove that there is an absolute constant c > 0 such that |A + λ·A| ≥ e^{c √(log |A|)} |A| for every finite A ⊆ ℝ and every transcendental λ ∈ ℝ, best possible up to the constant c by the construction of Konyagin and Łaba.

public theorem_wanted transcendentalDilateLowerBound_holds : transcendentalDilateLowerBound

Sources

• Problems presented at the workshop on recent trends in additive combinatorics, collected by E. Croot and V. F. Lev, AIM workshop problem list, Problem 4.8 (I. Łaba), https://aimath.org/WWN/additivecomb/additivecomb.pdf; resolution: D. Conlon and J. Lim, Sums of transcendental dilates, Bull. London Math. Soc. 55 (2023) 2400-2406, arXiv:2212.10128, abstract and main theorem, https://arxiv.org/abs/2212.10128.

Uses 1

Statement usesLower bound for sums of transcendental dilates

1 fully proved1 ready to prove

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