arXiv:2610.07471v1

Gaps theorems for linear forms with cubic coefficients

math.NTmath.DS11J1311J7137A17

Paper metadata

Original submission: October 5, 2026
Claimed MSC: 11J13, 11J71, 37A17
arXiv categories: math.NT, math.DS
Paper license: CC BY 4.0 ↗
Comments: 15 pages

Abstract

Let $1,α_1,α_2$ be a basis for a cubic number field $K\subset\mathbb{R}$ , and for $τ_1,τ_2\ge 1$ let $G(\mathbfα,\mathbfτ)$ denote the number of distinct gaps between the fractional parts of the numbers $m_1α_1+m_2α_2$ , with $m_1,m_2\in\mathbb{Z}$ , $0\le m_1<τ_1$ and $0\le m_2<τ_2$ . We show that $G(\mathbfα,\mathbfτ)$ is unbounded as $\mathbfτ$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,κ>0$ for which $G(\mathbfα,\mathbfτ)\ge c(\log\max\{τ_1,τ_2\})^κ$ for infinitely many $\mathbfτ\in\mathbb{N}^2$ .

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 24, 2026

01Statements3 reported findingsNo statement issues found

The unbounded-gap theorem for totally real cubic fields and the quantitative lower bound for cubic fields with a complex embedding follow under the stated basis hypotheses. The lattice correspondence, semigroup-density reduction, direct gap ladder, and effective rotation-time estimate retain the required signs, integrality, and parameter dependence.

Theorem 1No issue found

The diagonal-semigroup orbit produces arbitrarily many distinct gaps

PDF pp. 2–5 · Proposition 4 and Lemma 5 · arXiv:2610.07471v1

Shapira's cited Corollary 5.2 gives density of the full positive diagonal orbit for the specified totally real cubic basis. Lemma 5 validly removes the part with either dilation coordinate at most $T$ : the corresponding dual lattices contain $s e_i$ with $0<s\leq T$ , and that exceptional set is closed with empty interior. Proposition 4 from the cited Haynes–Marklof paper supplies $\lfloor\lambda/\epsilon\rfloor$ open parameter cells with values $\epsilon(j+a_j)$ and $|a_j|\leq 1/100$ . Choosing both dilation coordinates larger than $2/\delta$ puts an integer point in each scaled cell, and the resulting values are pairwise distinct. Letting $\epsilon\downarrow0$ proves the theorem.

Proposition 9No issue found

The primitive two-row coefficient lattice yields the asserted ladder of gaps

PDF pp. 8–10 · arXiv:2610.07471v1

The determinant estimate for every lattice point below the ladder is strictly less than $1$ because $4MN\beta<1$ and $C_1,C_2\geq3$ . The gcd-one hypothesis on the three two-by-two minors then identifies the generated rank-two subgroup with all integer points in its rational span. For the translated base point $k_0=(0,(j_0-1)|m_2|)$ , the window inequalities force every competing positive value below the top rung to be an integral combination with second coefficient $1$ and first coefficient at least $j_0$ . Thus the gap is exactly $j_0\beta+\beta'$ , giving at least $\min\{\lfloor C_1\rfloor,\lfloor C_2\rfloor\}$ distinct lengths.

Theorem 2No issue found

The complex-embedding construction gives the claimed logarithmic-power lower bound

PDF pp. 5–14 · Lemmas 6–13 and Proposition 11 · arXiv:2610.07471v1

Continued-fraction convergents produce primitive elements $\gamma$ with $Q=|\sigma(\gamma)|\to\infty$ , prescribed quadrant, and $Q\operatorname{Im}\sigma(\gamma)=O(1)$ . Multiplication by a fixed unit preserves the primitive coefficient plane, while rotation into the arc $A_\gamma$ gives the sign conditions, $\min(C_1,C_2)\gg Q$ , and $MN\beta<1/4$ . The quoted Baker–Wüstholz bound applies to the algebraic unit-circle number $\xi=\sigma(u)/\overline{\sigma(u)}$ , which is not a root of unity, and yields a hitting time $k\ll Q^{2(C+1)}$ . Proposition 11 then gives $G(\alpha,\tau)\gg Q$ and $\log|\tau|\ll Q^{2(C+1)}$ , hence the stated bound for infinitely many integer pairs $\tau$ .

02Proofs2 reported findingsNo proof issues found

The two main proof chains are complete. The critical external density and gap-value inputs match their cited statements, and the internal determinant, unit, continued-fraction, trigonometric, and hitting-time estimates close with the printed constants.

Sections 2–3No issue found in the reviewed proof

The lattice correspondence and tail-semigroup density argument close Theorem 1

PDF pp. 3–5 · arXiv:2610.07471v1

The specialization of the earlier gap identity has the correct determinant factor $\tau_1\tau_2$ . The dual-lattice vectors used to identify the excluded semigroup region are computed correctly, closedness follows after choosing convergent representatives and a constant integer subsequence, and a countable union of lower-dimensional rotation cosets proves empty interior. These facts supply exactly the tail density needed to use every open cell from Proposition 4.

Sections 4–6No issue found in the reviewed proof

The primitive-element construction and quantitative rotation argument close Theorem 2

PDF pp. 5–14 · arXiv:2610.07471v1

The trace-dual coordinate formulas and rank-one unit group give an irrational rotation. Lemma 8 controls the nearly real conjugate with a continued-fraction error of order $1/q_r$ , and Lemma 10 correctly preserves the gcd of the coefficient minors under the integral automorphism induced by $u^k$ . In Proposition 11 the error terms are uniformly one tenth of all four main terms once $k\gg\log Q$ ; its trigonometric estimates imply both ladder ratios and the strict product bound. Lemma 12's Dirichlet step gives an interval hit in at most $R(\Phi(R)^{-1}+1)$ iterates, and the Baker–Wüstholz lower bound supplies the displayed polynomial choice of $k$ .

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2610.07471v1
Authors listed: Alan Haynes
Audit date: October 24, 2026

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Gaps theorems for linear forms with cubic coefficients

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The diagonal-semigroup orbit produces arbitrarily many distinct gaps

The primitive two-row coefficient lattice yields the asserted ladder of gaps

The complex-embedding construction gives the claimed logarithmic-power lower bound

The lattice correspondence and tail-semigroup density argument close Theorem 1

The primitive-element construction and quantitative rotation argument close Theorem 2

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