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

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 $$N\le x,y,z,t \le N+cN^{12/13}.$$