Bipartite Diophantine tuples and their applications

TL;DR

The paper develops a unified toolkit for BD$_k(n)$-type bipartite Diophantine tuples, combining gap principles, sieve bounds, and additive combinatorics to derive sharp size bounds. It proves an anti-gap principle for BD$_2(n)$, then uses that, plus sieve methods, to obtain general upper bounds on the sizes of the two sets in a bipartite tuple, including conditional (ABC) refinements for small $A$. It further explores connections to variants such as Banks–Luca–Szalay and Kihel–Kihel, and extends the scope to multiplicative Hilbert cubes, including effective finite bounds when the shift is a square. The simultaneous Pell-equation framework in the final section underpins the simultaneous-diophantine-approximation aspects and yields explicit growth constraints important for these BD problems.

Abstract

This paper investigates bipartite variants of generalized Diophantine tuples and their applications. We generalize a result of Bugeaud--Dujella on a special family of bipartite Diophantine tuples and affirmatively resolve a related question posed by the second author. Additionally, we establish new connections between bipartite Diophantine tuples and several known variants of Diophantine tuples, including those introduced by Banks--Luca--Szalay and Kihel--Kihel.

Theorems & Definitions (43)

• Theorem 1.1: Y24
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Lemma 2.1: Y24
• Corollary 2.2