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

Abstract

Representative examples of our results are as follows. For any positive integer $N$ the equation $$x^3+y^3=z^3+t^3,x,y,z,t\in \mathbb{N},\{x,y\}\ne \{z,t\}$$ has no solutions satisfying $$N\le x,y,z,t<N+(\frac{38}{3}N+\frac{1297}{36}{)}^{1/2}+\frac{19}{6}.$$ 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 $$N\le x,y,z,t\le N+(\frac{38}{3}N+\frac{1297}{36}{)}^{1/2}+\frac{19}{6}.$$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation has a solution satisfying $$N\le x,y,z,t\le N+c{N}^{2/3}.$$ For any $\epsilon >0$ there exist infinitely many positive integers $N$ such that the equation has no solutions satisfying $$N\le x,y,z,t\le N+N^{4/7-\epsilon }.$$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation $$x^4+y^4=z^4+t^4,x,y,z,t\in \mathbb{N},\{x,y\}\ne \{z,t\},$$ has no solutions satisfying $$N\le x,y,z,t\le N+c{N}^{3/5}.$$ There is an absolute constant $c>0$ such that for any positive integer $N$ this equation has a solution satisfying $$N\le x,y,z,t\le N+c{N}^{12/13}.$$

Minor typographical corrections