Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples

TL;DR

This paper explores the multiplicative structure of shifted subgroups in finite fields, providing new bounds and results relevant to Diophantine tuples and multiplicative decompositions, advancing understanding in additive and multiplicative combinatorics.

Contribution

It offers new bounds on product sets within shifted subgroups, refines existing results on Diophantine tuples, and addresses conjectures on multiplicative decompositions over finite fields.

Findings

Bound on product sets in shifted subgroups related to subgroup size

Sharper upper bounds for the size of sets with specific product properties

Maximum size determination of generalized Diophantine tuples over finite fields

Abstract

In this paper, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup $G$ contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is $n$ less than a $k$-th power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress towards a conjecture of S\'{a}rk\"{o}zy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in \mathbb{F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in $\mathbb{F}_p$ non-trivially.