Split families of Thue equations

DefinitionSplit Thue recurrence family#✓fully proved

Let \(d \in \mathbb{N}\), let \(G_i(n)\) be integers for \(i, n \in \mathbb{N}\), and let \(\gamma_i \in \mathbb{R}\) and \(\delta_i \in \mathbb{C}\) for \(i \in \mathbb{N}\). Then \((d, G, \gamma, \delta)\) is a split Thue recurrence family when \(d \ge 3\) and there are finite sets \(R_i \subseteq \mathbb{C}\) and complex numbers \(c_i(r)\) (\(i \in \mathbb{N}\), \(r \in \mathbb{C}\)) such that: for every \(1 \le i \le d\), \(c_i(r) \ne 0\) for all \(r \in R_i\), \(G_i(n) = \sum_{r \in R_i} c_i(r) r^n\) for all \(n\), \(\gamma_i \in R_i\), and \(|r| < \gamma_i\) for all \(r \in R_i\) with \(r \ne \gamma_i\); \(\gamma_1 \ge 0\), \(\gamma\) is strictly increasing on \(\{1, \dots, d - 1\}\), and \(\gamma_{d-1} = \gamma_d\); \(c_{d-1}(\gamma_{d-1}) = c_d(\gamma_d)\); for \(i \in \{d - 1, d\}\), \(\delta_i \in R_i\), \(\delta_i \ne \gamma_i\), and \(|r| < |\delta_i|\) for all \(r \in R_i\) other than \(\gamma_i\) and \(\delta_i\); and \(|\delta_{d-1}| < |\delta_d| < \gamma_{d-2}\) and \(\gamma_{d-2}^2 < \gamma_d |\delta_d|\).

In words: each \(G_i\) is a simple linear recurrence with dominant root \(\gamma_i\), the sequences \(G_{d-1}\) and \(G_d\) share their dominant root and its coefficient, and \(\delta_{d-1}\), \(\delta_d\) are their second dominant roots.

public def IsSplitThueRecurrenceFamily (d : ℕ) (G : ℕ → ℕ → ℤ) (γ : ℕ → ℝ) (δ : ℕ → ℂ) :
    Prop :=
  3 ≤ d ∧ ∃ (R : ℕ → Finset ℂ) (c : ℕ → ℂ → ℂ),
    (∀ i ∈ Finset.Icc 1 d, (∀ r ∈ R i, c i r ≠ 0) ∧
      ∀ n, (G i n : ℂ) = ∑ r ∈ R i, c i r * r ^ n) ∧
    (∀ i ∈ Finset.Icc 1 d, ((γ i : ℝ) : ℂ) ∈ R i ∧ ∀ r ∈ R i, r ≠ γ i → ‖r‖ < γ i) ∧
    0 ≤ γ 1 ∧ StrictMonoOn γ (Set.Icc 1 (d - 1)) ∧ γ (d - 1) = γ d ∧
    c (d - 1) (γ (d - 1)) = c d (γ d) ∧
    (∀ i ∈ ({d - 1, d} : Finset ℕ), δ i ∈ R i ∧ δ i ≠ γ i ∧
      ∀ r ∈ R i, r ≠ γ i → r ≠ δ i → ‖r‖ < ‖δ i‖) ∧
    ‖δ (d - 1)‖ < ‖δ d‖ ∧ ‖δ d‖ < γ (d - 2) ∧ γ (d - 2) ^ 2 < γ d * ‖δ d‖

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Theorem 1 (conditions 1-3); dominant roots taken real and nonnegative as in the paper's standing assumption after Corollary 1.

Used by 7

Used byExample of a split Thue recurrence family · Determinants of logarithms in a split Thue family · Asymptotics of the roots of a split Thue family · Growth of the roots of a split Thue family · Hilgart's theorem on split families of Thue equations · Split Thue families eventually have only trivial solutions · A split Thue recurrence family has at least three sequences

TheoremA split Thue recurrence family has at least three sequences#✓fully proved

If \((d, G, \gamma, \delta)\) is a split Thue recurrence family, then \(d \ge 3\).

public theorem IsSplitThueRecurrenceFamily.three_le {d : ℕ} {G : ℕ → ℕ → ℤ} {γ : ℕ → ℝ}
    {δ : ℕ → ℂ} (h : IsSplitThueRecurrenceFamily d G γ δ) : 3 ≤ d

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Theorem 1 (the conditions mention γ (d - 2)).

Uses 1

Statement usesSplit Thue recurrence family

TheoremExample of a split Thue recurrence family#✓fully proved

Let \(G_1(n) = 3^n\), \(G_2(n) = 5^n + 1\) and \(G_i(n) = 5^n + 2^n\) for \(i \notin \{1, 2\}\); let \(\gamma_1 = 3\) and \(\gamma_i = 5\) for \(i \ne 1\); and let \(\delta_2 = 1\) and \(\delta_i = 2\) for \(i \ne 2\). Then \((3, G, \gamma, \delta)\) is a split Thue recurrence family: the dominant roots of \(G_1, G_2, G_3\) are \(3, 5, 5\) and the second dominant roots of \(G_2, G_3\) are \(1, 2\).

public theorem isSplitThueRecurrenceFamily_example :
    IsSplitThueRecurrenceFamily 3
      (fun i n => if i = 1 then 3 ^ n else if i = 2 then 5 ^ n + 1 else 5 ^ n + 2 ^ n)
      (fun i => if i = 1 then 3 else 5) (fun i => if i = 2 then 1 else 2)

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Theorem 1 (conditions 1-3).

Uses 1

Statement usesSplit Thue recurrence family

TheoremHilgart's theorem on split families of Thue equations#●ready to prove

Let \((d, G, \gamma, \delta)\) be a split Thue recurrence family. Then there is \(\kappa \in \mathbb{R}\) such that for every \(n \in \mathbb{N}\) and all integers \(x, y\) with \(|y| \ge 2\) and

\[(x - G_1(n) y) \cdots (x - G_d(n) y) - y^d = \pm 1\]

we have \(\log |x| \le \kappa\), \(\log |y| \le \kappa\) and \(n \le \kappa\) (where \(\log 0 = 0\)).

public theorem_wanted splitThue_recurrence_solutions_bounded {d : ℕ} {G : ℕ → ℕ → ℤ}
    {γ : ℕ → ℝ} {δ : ℕ → ℂ} (hF : IsSplitThueRecurrenceFamily d G γ δ) :
    ∃ κ : ℝ, ∀ (n : ℕ) (x y : ℤ), 2 ≤ |y| →
      ((∏ i ∈ Finset.Icc 1 d, (x - G i n * y)) - y ^ d = 1 ∨
        (∏ i ∈ Finset.Icc 1 d, (x - G i n * y)) - y ^ d = -1) →
      Real.log |(x : ℝ)| ≤ κ ∧ Real.log |(y : ℝ)| ≤ κ ∧ (n : ℝ) ≤ κ

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Theorem 1. The computability of κ is not recorded.

Uses 1

Statement usesSplit Thue recurrence family

TheoremSplit Thue families eventually have only trivial solutions#●ready to prove

Let \((d, G, \gamma, \delta)\) be a split Thue recurrence family. Then there is \(n_0\) such that for every \(n \ge n_0\) and all integers \(x, y\),

\[(x - G_1(n) y) \cdots (x - G_d(n) y) - y^d = \pm 1\]

holds if and only if \((x, y) = (\pm 1, 0)\) or, for some \(1 \le i \le d\), \((x, y) = (G_i(n), 1)\) or \((x, y) = (-G_i(n), -1)\).

public theorem_wanted splitThue_recurrence_solutions_eventually_trivial {d : ℕ}
    {G : ℕ → ℕ → ℤ} {γ : ℕ → ℝ} {δ : ℕ → ℂ} (hF : IsSplitThueRecurrenceFamily d G γ δ) :
    ∃ n₀ : ℕ, ∀ n ≥ n₀, ∀ x y : ℤ,
      ((∏ i ∈ Finset.Icc 1 d, (x - G i n * y)) - y ^ d = 1 ∨
        (∏ i ∈ Finset.Icc 1 d, (x - G i n * y)) - y ^ d = -1) ↔
      ((x = 1 ∨ x = -1) ∧ y = 0) ∨
        ∃ i ∈ Finset.Icc 1 d, (x = G i n ∧ y = 1) ∨ (x = -G i n ∧ y = -1)

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Corollary 1. The computability of n₀ is not recorded.

Uses 1

Statement usesSplit Thue recurrence family

TheoremAsymptotics of the roots of a split Thue family#●ready to prove

Let \((d, G, \gamma, \delta)\) be a split Thue recurrence family. Call reals \(\alpha_i(n)\) (\(i, n \in \mathbb{N}\)) an admissible labelling if there is \(N\) such that for all \(n \ge N\) and \(1 \le i \le d\), \(\prod_{k=1}^{d} (\alpha_i(n) - G_k(n)) = 1\) and \(|\alpha_i(n) - G_i(n)| < 1\). Then, first, there is \(N\) such that for all \(n \ge N\) every \(z \in \mathbb{C}\) with \(\prod_{k=1}^{d} (z - G_k(n)) = 1\) is real; second, an admissible labelling exists; third, for every admissible labelling \(\alpha\) and every \(1 \le i \le d\) there are \(C \in \mathbb{R}\) and \(N\) such that for all \(n \ge N\)

\[|(\alpha_i(n) - G_i(n)) \prod_{k \ne i} (G_i(n) - G_k(n)) - 1| \le C \Gamma_i^{-n},\]

where the product is over \(1 \le k \le d\) with \(k \ne i\), and \(\Gamma_i = \gamma_i^{i-1} (\prod_{k=i+1}^{d-2} \gamma_k) \gamma_d^2\) if \(i \le d - 2\) and \(\Gamma_i = \gamma_d^{d-2} |\delta_d|\) otherwise.

public theorem_wanted splitThue_recurrence_roots_asymptotic {d : ℕ} {G : ℕ → ℕ → ℤ}
    {γ : ℕ → ℝ} {δ : ℕ → ℂ} (hF : IsSplitThueRecurrenceFamily d G γ δ) :
    (∃ N : ℕ, ∀ n ≥ N, ∀ z : ℂ, ∏ k ∈ Finset.Icc 1 d, (z - G k n) = 1 → z.im = 0) ∧
    (∃ α : ℕ → ℕ → ℝ, ∃ N : ℕ, ∀ n ≥ N, ∀ i ∈ Finset.Icc 1 d,
      ∏ k ∈ Finset.Icc 1 d, (α n i - G k n) = 1 ∧ |α n i - G i n| < 1) ∧
    ∀ α : ℕ → ℕ → ℝ, (∃ N : ℕ, ∀ n ≥ N, ∀ i ∈ Finset.Icc 1 d,
      ∏ k ∈ Finset.Icc 1 d, (α n i - G k n) = 1 ∧ |α n i - G i n| < 1) →
      ∀ i ∈ Finset.Icc 1 d, ∃ (C : ℝ) (N : ℕ), ∀ n ≥ N,
        |(α n i - G i n) * ∏ k ∈ (Finset.Icc 1 d).erase i, ((G i n : ℝ) - G k n) - 1| ≤
          C * ((if i ≤ d - 2 then γ i ^ (i - 1) * (∏ k ∈ Finset.Icc (i + 1) (d - 2), γ k) *
            γ d ^ 2 else γ d ^ (d - 2) * ‖δ d‖)⁻¹) ^ n

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Lemma 1. The O-term is written as |(α⁽ⁱ⁾ - G i n) * ∏_{k ≠ i} (G i n - G k n) - 1| ≤ C * γ_ε⁻ⁿ; δ d enters through its modulus.

Uses 1

Statement usesSplit Thue recurrence family

TheoremGrowth of the roots of a split Thue family#●ready to prove

Let \((d, G, \gamma, \delta)\) be a split Thue recurrence family, and let \(\alpha_i(n)\) (\(i, n \in \mathbb{N}\)) be reals for which there is \(N\) such that for all \(n \ge N\) and \(1 \le i \le d\), \(\prod_{k=1}^{d} (\alpha_i(n) - G_k(n)) = 1\) and \(|\alpha_i(n) - G_i(n)| < 1\). Then there are \(c > 0\) and \(N'\) such that for all \(n \ge N'\) and \(1 \le i \le d\) we have \(|\alpha_i(n)| \ge c \gamma_i^n\), and for every \(1 \le j \le d\) with \(j \ne i\) both \(|\alpha_i(n) - \alpha_j(n)|\) and \(|\alpha_i(n) - G_j(n)|\) are at least \(c |\delta_d|^n\) if \(\{i, j\} = \{d - 1, d\}\), and at least \(c \max(\gamma_i, \gamma_j)^n\) otherwise.

public theorem_wanted splitThue_recurrence_roots_growth {d : ℕ} {G : ℕ → ℕ → ℤ}
    {γ : ℕ → ℝ} {δ : ℕ → ℂ} (hF : IsSplitThueRecurrenceFamily d G γ δ)
    (α : ℕ → ℕ → ℝ) (hα : ∃ N : ℕ, ∀ n ≥ N, ∀ i ∈ Finset.Icc 1 d,
      ∏ k ∈ Finset.Icc 1 d, (α n i - G k n) = 1 ∧ |α n i - G i n| < 1) :
    ∃ c > 0, ∃ N : ℕ, ∀ n ≥ N, ∀ i ∈ Finset.Icc 1 d,
      c * γ i ^ n ≤ |α n i| ∧ ∀ j ∈ Finset.Icc 1 d, j ≠ i →
        (({i, j} : Finset ℕ) = {d - 1, d} →
          c * ‖δ d‖ ^ n ≤ |α n i - α n j| ∧ c * ‖δ d‖ ^ n ≤ |α n i - G j n|) ∧
        (({i, j} : Finset ℕ) ≠ {d - 1, d} →
          c * max (γ i) (γ j) ^ n ≤ |α n i - α n j| ∧
            c * max (γ i) (γ j) ^ n ≤ |α n i - G j n|)

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Lemma 2, including its last sentence (the same bounds with α⁽ʲ⁾ replaced by G j n). Ω is the paper's lower bound |f n| ≥ c * g n for large n.

Uses 1

Statement usesSplit Thue recurrence family

TheoremDeterminants of logarithms in a split Thue family#●ready to prove

Let \((d, G, \gamma, \delta)\) be a split Thue recurrence family, let \(\alpha_i(n)\) (\(i, n \in \mathbb{N}\)) be reals for which there is \(N\) such that for all \(n \ge N\) and \(1 \le i \le d\), \(\prod_{l=1}^{d} (\alpha_i(n) - G_l(n)) = 1\) and \(|\alpha_i(n) - G_i(n)| < 1\), and let \(1 \le k \le d - 1\). Let \(B_k(n)\) be the \(k \times k\) matrix with entries \(\log |\alpha_i(n) - G_j(n)|\) for \(1 \le i, j \le k\), where \(\log 0 = 0\). Then there are \(c_1 > 0\), \(c_2 > 0\) and \(N'\) such that \(c_1 n^k \le |\det B_k(n)| \le c_2 n^k\) for all \(n \ge N'\).

public theorem_wanted splitThue_recurrence_logUnitDet_asymptotic {d : ℕ} {G : ℕ → ℕ → ℤ}
    {γ : ℕ → ℝ} {δ : ℕ → ℂ} (hF : IsSplitThueRecurrenceFamily d G γ δ)
    (α : ℕ → ℕ → ℝ) (hα : ∃ N : ℕ, ∀ n ≥ N, ∀ i ∈ Finset.Icc 1 d,
      ∏ k ∈ Finset.Icc 1 d, (α n i - G k n) = 1 ∧ |α n i - G i n| < 1) (k : ℕ) (hk : 1 ≤ k) (hkd : k
          ≤ d - 1) :
    ∃ c₁ > 0, ∃ c₂ > 0, ∃ N : ℕ, ∀ n ≥ N,
      c₁ * (n : ℝ) ^ k ≤
          |(Matrix.of fun a b : Fin k => Real.log (α n ((a : ℕ) + 1) - G ((b : ℕ) + 1) n)).det| ∧
        |(Matrix.of fun a b : Fin k => Real.log (α n ((a : ℕ) + 1) - G ((b : ℕ) + 1) n)).det| ≤
          c₂ * (n : ℝ) ^ k

Sources

• Tobias Hilgart, On Split Families of Thue Equations with Linear Recurrence Sequences as Factors, INTEGERS 25 (2025), Article A35, Lemma 3. Real.log is the logarithm of the absolute value; Θ is the paper's two-sided bound c₁ n ^ k ≤ |det B_k| ≤ c₂ n ^ k for large n.

Uses 1

Statement usesSplit Thue recurrence family

3 fully proved5 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.