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

Abstract

For all $α_1,α_2\in(1,2)$ with $1/α_1+1/α_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor{n_1^{α_1}}\rfloor+\lfloor{n_2^{α_2}}\rfloor$ is equal to $Γ(1+1/α_1)Γ(1+1/α_2)Γ(1/α_1+1/α_2)^{-1}N^{1/α_1+1/α_2-1} + o(N^{1/α_1+1/α_2-1})$ as $N\to\infty$, where $Γ$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when $α_1,α_2\in(1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.