A uniform metrical theorem in multiplicative Diophantine approximation

Journal article, 2026

For Lebesgue generic (x(1), x(2)) is an element of R-2, we investigate the distribution of small values of products q & centerdot; ||qx(1)||& centerdot;||qx(2)|| with q is an element of N, where ||& centerdot;|| denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of 1 <= q< T such that a<q & centerdot; ||qx(1) ||& centerdot;||qx(2) ||<= b and ||qx(1)|| , ||qx(2)|| <= c for given parameters a, b, c satisfying certain growth conditions.