N-continued fraction expansions

DefinitionDigit of the N-expansion map#✓fully proved

Let \(N\) be a natural number and \(\alpha\) a real number, and write \(I_\alpha = [\alpha, \alpha + 1]\). The digit of a real number \(x\) is \(d(x) = \lfloor N/x - \alpha \rfloor\), except that when \(N/\alpha - \alpha\) is an integer, the digit of \(\alpha\) itself is \(d(\alpha) = N/\alpha - \alpha - 1\).

With this choice the \(N\)-expansion map \(T_\alpha(x) = N/x - d(x)\) sends \(\alpha\) to \(\alpha + 1\) in the integer case, every other point of \(I_\alpha\) lands in \([\alpha, \alpha + 1)\), and no cylinder is a single point.

public noncomputable def nExpansionDigit (N : ℕ) (α x : ℝ) : ℤ :=
  if x = α ∧ ∃ k : ℤ, (N : ℝ) / α - α = k then ⌊(N : ℝ) / x - α⌋ - 1 else ⌊(N : ℝ) / x - α⌋

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 1.

Used by 7

Used byThe ideal periodic point is the least point of its orbit · The ideal periodic points decrease · The digit away from the left endpoint · The \(N\)-expansion map · The N-expansion map away from the left endpoint · The left endpoint goes to the right endpoint when \(N/\alpha - \alpha\) is an integer · Orbits with the same digits alternate in order

DefinitionThe \(N\)-expansion map#✓fully proved

For a natural number \(N\) and a real number \(\alpha\), the \(N\)-expansion map is \(T_\alpha(x) = N/x - d_\alpha(x)\) for real \(x\), where \(d_\alpha(x)\) is the \(N\)-expansion digit: \(d_\alpha(x) = \lfloor N/x - \alpha \rfloor\), except that \(d_\alpha(\alpha) = \lfloor N/\alpha - \alpha \rfloor - 1\) when \(N/\alpha - \alpha\) is an integer. The map is meant to act on the interval \(I_\alpha = [\alpha, \alpha + 1]\), but it is defined for every real \(x\), with the convention \(N/0 = 0\).

public noncomputable def nExpansionMap (N : ℕ) (α x : ℝ) : ℝ :=
  (N : ℝ) / x - nExpansionDigit N α x

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 1.

Uses 1 · Used by 5

Statement usesDigit of the N-expansion map

Used byThe ideal periodic point is the least point of its orbit · The ideal periodic points decrease · The N-expansion map away from the left endpoint · The left endpoint goes to the right endpoint when \(N/\alpha - \alpha\) is an integer · Orbits with the same digits alternate in order

TheoremThe digit away from the left endpoint#✓fully proved

Let \(N\) be a natural number and let \(\alpha\) and \(x\) be real numbers with \(x \ne \alpha\). Then the digit of \(x\) for the \(N\)-expansion map on \([\alpha, \alpha + 1]\) is \(\lfloor N/x - \alpha \rfloor\).

public theorem nExpansionDigit_of_ne {N : ℕ} {α x : ℝ} (hx : x ≠ α) :
    nExpansionDigit N α x = ⌊(N : ℝ) / x - α⌋

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 1.

Uses 1 · Used by 1

Statement usesDigit of the N-expansion map

Used byThe N-expansion map away from the left endpoint

TheoremThe N-expansion map away from the left endpoint#✓fully proved

Let \(N\) be a natural number and let \(\alpha\) and \(x\) be real numbers with \(x \ne \alpha\). Then the \(N\)-expansion map \(T_\alpha\) satisfies \(\alpha \le T_\alpha(x) < \alpha + 1\).

Indeed \(T_\alpha(x) = \alpha + \{N/x - \alpha\}\), where \(\{\cdot\}\) denotes the fractional part.

public theorem nExpansionMap_mem_Ico_of_ne {N : ℕ} {α x : ℝ} (hx : x ≠ α) :
    nExpansionMap N α x ∈ Set.Ico α (α + 1)

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 1.

Uses 3

Statement usesThe \(N\)-expansion map

Proof usesDigit of the N-expansion map · The digit away from the left endpoint

TheoremThe left endpoint goes to the right endpoint when \(N/\alpha - \alpha\) is an integer#✓fully proved

Let \(N\) be a natural number and \(\alpha\) a real number such that \(N/\alpha - \alpha\) is an integer, with the convention \(N/0 = 0\). Then the \(N\)-expansion map \(T_\alpha\) satisfies \(T_\alpha(\alpha) = \alpha + 1\).

public theorem nExpansionMap_self_of_int {N : ℕ} {α : ℝ} {k : ℤ} (hk : (N : ℝ) / α - α = k) :
    nExpansionMap N α α = α + 1

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 1.

Uses 2

Statement usesThe \(N\)-expansion map

Proof usesDigit of the N-expansion map

DefinitionIdeal period words#✓fully proved

For \(n \ge 1\), the ideal period word \(d_n\) is a finite list of blocks of partial quotients, each block being \(a = 2, 1, 1\) or \(b = 2, 2\). Writing lists of blocks by juxtaposition, with commas between parts, \(d_1 = [ab]\), \(d_2 = [abb]\), \(d_3 = [abb, ab, ab]\) and \(d_4 = [abb, ab, ab, abb, abb]\); for \(n \ge 5\), if \(d_{n-1} = [j, ab, ab]\) for a list \(j\) then \(d_n = [j, ab, ab, j, abb]\), and if \(d_{n-1} = [j, abb]\) then \(d_n = [j, abb, j, ab, ab]\). The digit string of \(d_n\) is the concatenation of its blocks. For \(n = 0\) the list is empty, a placeholder.

public def idealPeriodWord : ℕ → List (List ℕ)
  | 0 => []
  | 1 => [[2, 1, 1], [2, 2]]
  | 2 => [[2, 1, 1], [2, 2], [2, 2]]
  | 3 => [[2, 1, 1], [2, 2], [2, 2], [2, 1, 1], [2, 2], [2, 1, 1], [2, 2]]
  | 4 => [[2, 1, 1], [2, 2], [2, 2], [2, 1, 1], [2, 2], [2, 1, 1], [2, 2], [2, 1, 1], [2, 2],
      [2, 2], [2, 1, 1], [2, 2], [2, 2]]
  | n + 5 =>
    let w := idealPeriodWord (n + 4)
    if w.drop (w.length - 4) = [[2, 1, 1], [2, 2], [2, 1, 1], [2, 2]] then
      w ++ w.take (w.length - 4) ++ [[2, 1, 1], [2, 2], [2, 2]]
    else if w.drop (w.length - 3) = [[2, 1, 1], [2, 2], [2, 2]] then
      w ++ w.take (w.length - 3) ++ [[2, 1, 1], [2, 2], [2, 1, 1], [2, 2]]
    else w

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 7.

Used by 6

Used byBlock counts of the ideal period words · The fifth ideal period word · Length of the ideal period words · Lengths of the first ideal period words · The ideal periodic point is the least point of its orbit · The ideal periodic points decrease

TheoremThe fifth ideal period word#✓fully proved

Writing \(a\) for the block \(2, 1, 1\) and \(b\) for the block \(2, 2\) of partial quotients, the ideal period word \(d_5\) is \(d_4\) followed by the blocks \(abb\,ab\,ab\,abb\,ab\,ab\), in this order.

This is the second recursion rule applied to \(d_4 = [j, abb]\) with \(j = [abb, ab, ab, abb]\).

public theorem idealPeriodWord_five :
    idealPeriodWord 5 = idealPeriodWord 4 ++
      [[2, 1, 1], [2, 2], [2, 2], [2, 1, 1], [2, 2], [2, 1, 1], [2, 2], [2, 1, 1], [2, 2], [2, 2],
        [2, 1, 1], [2, 2], [2, 1, 1], [2, 2]]

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 7.

Uses 1

Statement usesIdeal period words

TheoremLengths of the first ideal period words#✓fully proved

The digit strings of the ideal period words \(d_1, d_2, d_3, d_4, d_5\) have lengths \(5, 7, 17, 31, 65\) respectively.

public theorem idealPeriodWord_flatten_length_small :
    ((List.range 5).map fun n => (idealPeriodWord (n + 1)).flatten.length) = [5, 7, 17, 31, 65]

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Definition 7.

Uses 1

Statement usesIdeal period words

TheoremOrbits with the same digits alternate in order#●ready to prove

Let \(N\) be a positive integer and let \(0 < \alpha < \alpha'\) be real numbers. Write \(T_\alpha\) for the \(N\)-expansion map and \(d_\alpha\) for the \(N\)-expansion digit with parameter \(\alpha\), and let \(m\) be a natural number. If the orbits \(\alpha_j = T_\alpha^j(\alpha)\) and \(\alpha'_j = T_{\alpha'}^j(\alpha')\) satisfy \(d_\alpha(\alpha_j) = d_{\alpha'}(\alpha'_j)\) for all \(j \in \{0, \dots, m\}\), then \(\alpha'_m < \alpha_m\) if and only if \(m\) is odd. Likewise, if the orbits \(\beta_j = T_\alpha^j(\alpha + 1)\) and \(\beta'_j = T_{\alpha'}^j(\alpha' + 1)\) satisfy \(d_\alpha(\beta_j) = d_{\alpha'}(\beta'_j)\) for all \(j \in \{0, \dots, m\}\), then \(\beta'_m < \beta_m\) if and only if \(m\) is odd.

The paper assumes \(\alpha\) and \(\alpha'\) in Phase I (so in particular \(\alpha' \le \sqrt N - 1\)); for \(N > 0\) neither assumption is needed (the orbits stay in \([\alpha, \alpha + 1]\) and \([\alpha', \alpha' + 1]\), where \(x \mapsto N/x\) is strictly decreasing), so both are replaced by \(N > 0\). For \(N = 0\) the claim fails (\(\alpha = 1/5\), \(\alpha' = 3/10\), \(m = 1\)).

public theorem_wanted nExpansion_orbit_gt_iff_odd {N : ℕ} (hN : 0 < N) {α α' : ℝ} (hα : 0 < α)
    (hαα' : α < α') (m : ℕ) :
    ((∀ j ≤ m, nExpansionDigit N α ((nExpansionMap N α)^[j] α) =
        nExpansionDigit N α' ((nExpansionMap N α')^[j] α')) →
      ((nExpansionMap N α')^[m] α' < (nExpansionMap N α)^[m] α ↔ Odd m)) ∧
    ((∀ j ≤ m, nExpansionDigit N α ((nExpansionMap N α)^[j] (α + 1)) =
        nExpansionDigit N α' ((nExpansionMap N α')^[j] (α' + 1))) →
      ((nExpansionMap N α')^[m] (α' + 1) < (nExpansionMap N α)^[m] (α + 1) ↔ Odd m))

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Lemma 2.

Uses 2

Statement usesDigit of the N-expansion map · The \(N\)-expansion map

TheoremThe ideal periodic points decrease#●ready to prove

Let \(N\) be a natural number, let \(\alpha\) be a real number with \(0 < \alpha \le \sqrt N - 1\), and let \(T_\alpha\) be the \(N\)-expansion map on \(I_\alpha = [\alpha, \alpha + 1]\). For \(m \ge 1\) write \(|d_m|\) for the length of the digit string of the ideal period word \(d_m\). Let \(k, \ell \ge 1\) and let \(x, y \in I_\alpha\) be such that for every \(i \ge 0\) the digit of \(T_\alpha^i(x)\) is entry \(i \bmod |d_k|\) (counting from \(0\)) of the digit string of \(d_k\), and the digit of \(T_\alpha^i(y)\) is entry \(i \bmod |d_\ell|\) of the digit string of \(d_\ell\); that is, the expansions of \(x\) and \(y\) are purely periodic with period words \(d_k\) and \(d_\ell\). Then \(x < y\) if and only if \(\ell < k\).

The source works in Phase I; only the assumption \(0 < \alpha \le \sqrt N - 1\) is kept here, which makes \(T_\alpha\) expanding, so that a digit sequence determines its point.

public theorem_wanted idealPeriodWord_point_lt_iff {N : ℕ} {α : ℝ} (hα : 0 < α)
    (hαN : α ≤ Real.sqrt N - 1) {k ℓ : ℕ} (hk : 1 ≤ k) (hℓ : 1 ≤ ℓ) {x y : ℝ}
    (hx : x ∈ Set.Icc α (α + 1)) (hy : y ∈ Set.Icc α (α + 1))
    (hxd : ∀ i, nExpansionDigit N α ((nExpansionMap N α)^[i] x) =
      ((idealPeriodWord k).flatten.getD (i % (idealPeriodWord k).flatten.length) 0 : ℤ))
    (hyd : ∀ i, nExpansionDigit N α ((nExpansionMap N α)^[i] y) =
      ((idealPeriodWord ℓ).flatten.getD (i % (idealPeriodWord ℓ).flatten.length) 0 : ℤ)) :
    x < y ↔ ℓ < k

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Lemma 4.

Uses 3

Statement usesIdeal period words · Digit of the N-expansion map · The \(N\)-expansion map

TheoremThe ideal periodic point is the least point of its orbit#●ready to prove

Let \(N\) be a natural number, let \(\alpha\) be a real number with \(0 < \alpha \le \sqrt N - 1\), and let \(T_\alpha\) be the \(N\)-expansion map on \(I_\alpha = [\alpha, \alpha + 1]\). Let \(n \ge 1\), write \(|d_n|\) for the length of the digit string of the ideal period word \(d_n\), and let \(x \in I_\alpha\) be such that for every \(i \ge 0\) the digit of \(T_\alpha^i(x)\) is entry \(i \bmod |d_n|\) (counting from \(0\)) of that digit string, so that the expansion of \(x\) is purely periodic with period word \(d_n\). Then \(x \le T_\alpha^j(x)\) for every \(j \ge 0\).

The source assumes Phase I; only the assumption \(0 < \alpha \le \sqrt N - 1\) is kept here.

public theorem_wanted idealPeriodWord_point_le_orbit {N : ℕ} {α : ℝ} (hα : 0 < α)
    (hαN : α ≤ Real.sqrt N - 1) {n : ℕ} (hn : 1 ≤ n) {x : ℝ} (hx : x ∈ Set.Icc α (α + 1))
    (hxd : ∀ i, nExpansionDigit N α ((nExpansionMap N α)^[i] x) =
      ((idealPeriodWord n).flatten.getD (i % (idealPeriodWord n).flatten.length) 0 : ℤ))
    (j : ℕ) :
    x ≤ (nExpansionMap N α)^[j] x

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Lemma 5.

Uses 3

Statement usesIdeal period words · Digit of the N-expansion map · The \(N\)-expansion map

TheoremLength of the ideal period words#●ready to prove

For every \(n \ge 1\), the digit string of the ideal period word \(d_n\) (the concatenation of its blocks of partial quotients) has length \(|d_n| = 2^{n+1} - (-1)^n\).

The source derives this from the recursion \(|d_n| = 2|d_{n-1}| - 3(-1)^n\).

public theorem_wanted idealPeriodWord_flatten_length {n : ℕ} (hn : 1 ≤ n) :
    ((idealPeriodWord n).flatten.length : ℤ) = 2 ^ (n + 1) - (-1) ^ n

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Lemma 7.

Uses 1

Statement usesIdeal period words

TheoremBlock counts of the ideal period words#●ready to prove

For every \(n \ge 1\), the ideal period word \(d_n\) contains exactly \(2^{n-1}\) blocks \(2, 2\), and the number \(c_n\) of its blocks \(2, 1, 1\) satisfies \(3 c_n = 2^n - (-1)^n\); that is, \(c_n = (2^n + 1)/3\) for odd \(n\) and \(c_n = (2^n - 1)/3\) for even \(n\).

In this lemma the source names the blocks \(a = 2, 2\) and \(b = 2, 1, 1\), the reverse of its Definition 7; the counts are stated by block, so the naming does not matter.

public theorem_wanted idealPeriodWord_count_blocks {n : ℕ} (hn : 1 ≤ n) :
    (idealPeriodWord n).count [2, 2] = 2 ^ (n - 1) ∧
      3 * ((idealPeriodWord n).count [2, 1, 1] : ℤ) = 2 ^ n - (-1) ^ n

Sources

• Jaap de Jonge, Intervals of N-Expansions with Many Gaps, INTEGERS 26 (2026), Article A75, Lemma 8.

Uses 1

Statement usesIdeal period words

8 fully proved5 ready to prove

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