Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average (Q2519155)

Let \(a,m,Y,Z\) be integers, \(\chi\) be a set of integers and \[ M_{a,m}(\chi;Y,Z)=\sharp\{(x,y)\in \chi\times[Z+1,Z+Y]: \gcd(x,m)=1,\,a/x\equiv y\pmod m]\}\,, \] \[ N_{a,m}(\chi;Y,Z)=\sharp\{(x,y)\in \chi\times[Z+1,Z+Y]: \gcd(x,\,m)=1,\,ax\equiv y\pmod m]\}\,, \] where the inversion is always taken modulo \(m\). The author extends some results of \textit{M. Z. Garaev} and \textit{A. A. Karatsuba} [``The representation of residue classes by products of small integers'', Proc. Edinb. Math. Soc., II. Ser. 50, No. 2, 363--375 (2007; Zbl 1197.11003)]. He show that if \(X,Y\geq m^{1/2+\varepsilon}\) and if \(\chi\) is sufficiently massive subset of the interval \([-X,X]\) then \(M_{a,m}(\chi;Y,Z)\) and \(N_{a,m}(\chi;Y,Z)\) are close to their expected average values for all but \(o(m)\) values of \(a=1,...,m\). Combining the obtained bounds of \(M_{a,m}(\chi;Y,Z)\) with result of \textit{H. Niederreiter} [``The distribution of values of Kloosterman sums'', Arch. Math. 56, No. 3, 270--277 (1991; Zbl 0752.11055)] the author shows that on average over \(r\) and \(s\) and ranging over relatively short intervals \(|r|\leq R,\,|s|\leq S\), the Sato-Tate conjecture holds on average and the sum \[ \Pi_{\alpha,\beta}(R,S,T)=\frac{1}{4RS} \sum\limits_{0<|r|\leq R} \sum\limits_{0<|s|\leq S}\pi_{r,s}(\alpha,\beta,T) \] satisfies \[ \Pi_{\alpha,\beta}(R,\,S,\,T)\sim\mu_{ST}(\alpha,\beta)\pi(T). \] Over larger intervals the dispersion \[ \Delta_{\alpha,\beta}(R,S,T)=\frac{1}{4RS} \sum\limits_{0<|r|\leq R} \sum\limits_{0<|s|\leq S} (\pi_{r,s}(\alpha,\beta,T)-\mu_{ST}(\alpha,\beta)\pi(T))^2 \] is estimated as \[ \max_{0\leq \alpha<\beta\leq \pi} \Delta_{\alpha,\beta}(R,S,T) \ll T^{3/4}+R^{-1/2}S^{-1/2}T^{3+o(1)}. \] The obtained results for \(\Pi_{\alpha,\beta}(R,S,T)\) and \(\Delta_{\alpha,\beta}(R,S,T)\) are nontrivial for \(RS\geq T^{1+\varepsilon}\) and \(RS\geq T^{2+\varepsilon}\), respectively.