Multiplicatively dependent integer vectors on a hyperplane

TL;DR

This work studies the distribution of multiplicatively dependent vectors on a fixed hyperplane in $\mathbb{G}_m^n$, counting those with coordinate height bounded by $H$. The authors combine determinant-method bounds (Bombieri–Pila-type results for curves) with lattice-volume techniques (Davenport-type shifted-lattice counts) to derive asymptotics and upper bounds for $S_n(H,J;\bm{\alpha})$ across cases determined by the hyperplane’s nonzero coordinate count $k$. For $k\ge 3$, they obtain a main term $C_{\bm{\alpha};J}H^{n-2}$ (and explicit constants in many subcases), while treating $k\le 4$ via refined rank-based decompositions, and they also analyze large rank and positive-coordinate variants. A key component is translating point-counts into volumes of hyperplane sections of boxes, enabling precise volume computations that yield the main-term coefficients; the work also provides a correction to a prior result (PSSS) and discusses potential extensions to algebraic integers and more general varieties. The results significantly advance the arithmetic-statistical understanding of multiplicatively dependent vectors on affine varieties and offer tools applicable to related distribution problems in number theory.

Abstract

We establish several asymptotic formulae and upper bounds for the count of multiplicatively dependent integer vectors that lie on a fixed hyperplane and have bounded height. This work constitutes a direct extension of the results obtained by Pappalardi, Sha, Shparlinski, and Stewart.

Theorems & Definitions (48)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof