Diophantine tuples and multiplicative structure of shifted multiplicative subgroups

Abstract

We provide a substantial improvement on a recent result by Dixit, Kim, and Murty on the upper bound of $M_k(n)$, the largest size of a generalized Diophantine tuple with property $D_k(n)$, that is, each pairwise product is $n$ less than a $k$-th power. In particular, we show $M_k(n)=o(\log n)$ for a specially chosen sequence of $k$ and $n$ tending to infinity, breaking the $\log n$ barrier unconditionally. One innovation of our proof is a novel combination of Stepanov's method and Gallagher's larger sieve. One main ingredient in our proof is a 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 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.Comment: 48 pages, 1 figur