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

Abstract Representative examples of our results are as follows. For any positive integer $$N$$ N the equation $$x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\}$$ x 3 + y 3 = z 3 + t 3 , x , y , z , t ∈ N , { x , y } ≠ { z , t } has no solutions satisfying $$N\le x,y,z,t < N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}$$ N ≤ x , y , z , t < N + ( 38 3 N + 1297 36 ) 1 / 2 + 19 6 . The strict inequality " $$<$$ < " cannot be substituted by " $$\le$$ ≤ ". There is an absolute constant $$c>0$$ c > 0 such that for any positive integer $$N$$ N the equation (0.1) has a solution satisfying $$N\le x,y,z,t \le N+cN^{2/3}$$ N ≤ x , y , z , t ≤ N + c N 2 / 3 . For any $$\varepsilon>0$$ ε > 0 there exist infinitely many positive integers $$N$$ N