On Sidon sets with squares, cubes and quartics in short intervals

TL;DR

The paper investigates Sidon properties for sets of polynomial values in short intervals, focusing on squares, cubes, and fourth powers. It develops a blend of parity analysis, Euler–Binet parametric constructions, and Pell-type diophantine equations to both exclude and construct nontrivial equal-sum representations within tight intervals, yielding sharp thresholds and infinite families. The main contributions are precise interval-length bounds ensuring Sidon-ness (and their sharpness) for squares, cubes, and fourth powers, together with explicit constructions showing when nontrivial equal-sum representations must exist. These results advance the understanding of Sidon sets among polynomial sequences and have implications for related harmonic analysis questions on trigonometric polynomials with polynomial frequencies.

Abstract

Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t < N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ The strict inequality ``$<$" can not be substituted by ``$\le$", that is, there exist infinitely many positive integers $N$ such that the equation has a solution with $$ N\le x,y,z,t \le N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{2/3}. $$ For any $\varepsilon>0$ there exist infinitely many positive integers $N$ such that the equation has no solutions satisfying $$ N\le x,y,z,t \le N+N^{4/7-\varepsilon}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation $$ x^4+y^4=z^4+t^4,\quad x,y,z,t\in\mathbb{N}, \quad \{x,y\}\not=\{z,t\}, $$ has no solutions satisfying $$ N\le x,y,z,t \le N+cN^{3/5}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ this equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{12/13}. $$

Theorems & Definitions (9)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Corollary 2
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7