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

For all α₁,α₂∈(1,2) with 1/α₁+1/α₂>5/3, we show that the number of pairs (n₁,n₂) of positive integers with N=n₁α₁+n₂α₂ is equal to Γ(1+1/α₁)Γ(1+1/α₂)Γ(1/α₁+1/α₂)⁻¹N1/α₁+1/α₂-1 + o(N1/α₁+1/α₂-1) as N→∞, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when α₁,α₂∈(1,2) lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.