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 $\Gamma (1+1/{\alpha }_1)\Gamma (1+1/{\alpha }_2)\Gamma {(1/{\alpha }_1+1/{\alpha }_2)}^{-1}{N}^{1/{\alpha }_1+1/{\alpha }_2-1}+o\left(N^{1/{\alpha }_1+1/{\alpha }_2-1}\right)$ as $N\to \infty$, where Γ 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.