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

Abstract

For all $\alpha_1,\alpha_2\in(1,2)$ with $1/\alpha_1+1/\alpha_2>5/3$, 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 $\Gamma(1+1/\alpha_1)\Gamma(1+1/\alpha_2)\Gamma(1/\alpha_1+1/\alpha_2)^{-1}N^{1/\alpha_1+1/\alpha_2-1} + o(N^{1/\alpha_1+1/\alpha_2-1})$ as $N\to\infty$, where $\Gamma$ 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.