Diophantine tuples and product sets in shifted powers
Ernie Croot, Chi Hoi Yip
TL;DR
This work advances the theory of Diophantine tuples by studying robust generalizations $D_k(n)$ and $D_{\\le d}(n)$, along with their infinite-power variant, and by linking these number-theoretic objects to product sets in shifted powers. The authors develop a novel synthesis of sieve methods, Diophantine approximation (via linear forms in logarithms), and extremal graph theory, producing unconditional upper bounds for $M_{\\le d}(n)$ and $M_{\\le\infty}(n)$, as well as improved bounds for bipartite tuples. They extend finite-field models to create verifiable bounds and derive uniform-dichotomy results for $A,B$ under a generalized BD$_k(n)$ framework. The paper also delivers conditional results under major conjectures (ABC, Uniformity CHM, and Lander–Parkin–Selfridge), including explicit constant bounds for large $k$ and tight bounds for $M_{\\le d}(n)$ and $ ilde f(x)$, thereby connecting these Diophantine questions to broader conjectural landscapes. Overall, the results sharpen previous work by BDHL and Yip, and illuminate the structure of product sets within shifted power sets via a rigorous, multifaceted approach.
Abstract
Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes--Dujella--Hajdu--Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory.