Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers

For all ${\alpha }_1,{\alpha }_2\in (1,2)$ with $1\mathrm{/}{\alpha }_1+1\mathrm{/}{\alpha }_2>5\mathrm{/}3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor n_1^{{\alpha }_1}\rfloor +\lfloor n_2^{{\alpha }_2}\rfloor$ is equal to $\mathrm{\Gamma }(1+1\mathrm{/}{\alpha }_1)\mathrm{\Gamma }(1+1\mathrm{/}{\alpha }_2)\mathrm{\Gamma }(1\mathrm{/}{\alpha }_1+1\mathrm{/}{\alpha }_2{)}^{-1}{N}^{1\mathrm{/}{\alpha }_1+1\mathrm{/}{\alpha }_2-1}+o(N^{1\mathrm{/}{\alpha }_1+1\mathrm{/}{\alpha }_2-1})$ as $N\to \mathrm{\infty }$, where $\mathrm{\Gamma }$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when ${\alpha }_1,{\alpha }_2\in (1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.