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