Improved upper bounds on Diophantine tuples with the property $D(n)$

TL;DR

The paper addresses the size of Diophantine tuples with the property $D(n)$ by improving upper bounds on the intermediate-element contribution. The authors refine the standard decomposition into $A_n,B_n,C_n$ with a tunable parameter $\epsilon$, introducing $A_n^{(\epsilon)}$ and $B_n^{(\epsilon)}$ and proving $A_n^{(\epsilon)}=O(\log \tfrac{1}{\epsilon})$ and $B_n^{(\epsilon)}\le 0.631\epsilon\log|n|+O(1)$. This yields $B_n\le A_n^{(\epsilon)}+B_n^{(\epsilon)}$ and $M_n\le A_n^{(\epsilon)}+B_n^{(\epsilon)}+C_n$, and with $\epsilon=\dfrac{\log\log|n|}{\log|n|}$, the paper derives $B_n=O(\log\log|n|)$ and $M_n\le 2\log|n|+O\left(\dfrac{\log|n|}{(\log\log|n|)^2}\right)$, improving the Becker–Murty bound. The core technique combines a gap principle, derived from key lemmas in D02, with a careful analysis of exponential growth for large elements and a recursive, Fibonacci-type bound to limit the number of large intermediate elements. These results support the conjectured uniform boundedness of $M_n$ and sharpen the asymptotic dependence on $|n|$ for the maximal $D(n)$-tuple size. The methods also illustrate how refined separations of element sizes can lead to markedly tighter global bounds.

Abstract

Let $n$ be a non-zero integer. A set $S$ of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. In this notes, we show the contribution of intermediate elements is $O(\log \log |n|)$, improving a result by Dujella. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$, improving the best-known upper bound on $M_n$ by Becker and Murty.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: D02
• Lemma 2.2: D02
• Lemma 2.3: D02
• Corollary 2.4
• proof
• proof : Proof of Theorem \ref{['thm:main2']}