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 $$x^{3+y^{3=z^{3+t^3,}}} \quad x,y,z,t\in \mathbb{N}, \quad {x,y}\not={z,t}$$ 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 $N\le x,y,z,t \le N+cN^{12/13}.$