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

Seoyoung Kim, Chi Hoi Yip, Semin Yoo

TL;DR

The paper investigates how a nontrivial shift of a finite-field multiplicative subgroup retains multiplicative structure, linking additive combinatorics and Diophantine problems. Central is a Stepanov-type bound that constrains product sets when AB+λ lies in the d-th power subgroup, yielding sharper Vinogradov-type estimates and driving Diophantine-tuple and Sárközy-type decomposition results. The authors develop a unified framework combining Stepanov’s method and Gallagher’s sieve to obtain new upper bounds for generalized Diophantine tuples over integers, exact sizes for several Diophantine-tuple configurations over finite fields, and progress on multiplicative decompositions of shifted subgroups. They also connect to the Paley graph conjecture, showing conditional improvements and demonstrating that many of the key bounds hold unconditionally. Collectively, the results yield uniform, sharp bounds across finite fields and integers, with implications for Diophantine equations, additive combinatorics, and multiplicative decompositions.

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 Sárközy 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.

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

TL;DR

The paper investigates how a nontrivial shift of a finite-field multiplicative subgroup retains multiplicative structure, linking additive combinatorics and Diophantine problems. Central is a Stepanov-type bound that constrains product sets when AB+λ lies in the d-th power subgroup, yielding sharper Vinogradov-type estimates and driving Diophantine-tuple and Sárközy-type decomposition results. The authors develop a unified framework combining Stepanov’s method and Gallagher’s sieve to obtain new upper bounds for generalized Diophantine tuples over integers, exact sizes for several Diophantine-tuple configurations over finite fields, and progress on multiplicative decompositions of shifted subgroups. They also connect to the Paley graph conjecture, showing conditional improvements and demonstrating that many of the key bounds hold unconditionally. Collectively, the results yield uniform, sharp bounds across finite fields and integers, with implications for Diophantine equations, additive combinatorics, and multiplicative decompositions.

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 contains a product set , then is essentially bounded by , refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of , the largest size of a set such that each pairwise product of its elements is less than a -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 Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes , the set cannot be decomposed as the product of two sets in non-trivially.
Paper Structure (24 sections, 37 theorems, 150 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 24 sections, 37 theorems, 150 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Diophantine tuples over integers
  3. Diophantine tuples over finite fields
  4. Multiplicative decompositions of shifted multiplicative subgroups
  5. Background
  6. Stepanov's method
  7. Gallagher's larger sieve inequality
  8. An effective estimate for $M_k(n, \frac{k}{k-2})$
  9. Implications of the Paley graph conjecture
  10. Preliminary estimations for product sets in shifted multiplicative subgroups
  11. Character sum estimate and the square root upper bound
  12. Estimates on and if
  13. Products and restricted products in shifted multiplicative subgroups
  14. Product set in a shifted multiplicative subgroup
  15. Restricted product set in a shifted multiplicative subgroup
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $d \mid (q-1)$ with $d \ge 2$. Let $S_d=\{x^d: x \in {\mathbb F}_q^*\}$. Let $A,B \subset {\mathbb F}_q^*$ and $\lambda \in {\mathbb F}_q^*$ with $|A|, |B| \geq 2$. Assume further that $\binom{|A|-1+\frac{q-1}{d}}{|A|} \not \equiv 0 \pmod p$ if $q \neq p$. If $AB+\lambda \subset S_d \cup \{0\}$, Moreover, when $\lambda \in S_d$, we have a stronger upper bound:

Figures (1)

  • Figure A.1: Comparison between the new bound $\nu_k$ and the bound $3\phi(k)$ in DKM22 when $2 \le k \le 1000$. The black dots denote $3\phi(k)$, and the blue dots denote $\nu_k$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4: Trivial upper bound
  • Theorem 1.5
  • Theorem 1.6
  • Example 1.7
  • Conjecture 1.8: Sárközy
  • Conjecture 1.9
  • Corollary 1.10
  • ...and 68 more