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

Abstract: 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 solution 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 solution 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 solution 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}. $ The talk is based on a joint paper of the speaker with M.~Z.~Garaev and F.~M.~Garayev.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)