Integral approximation sequences

Abstract

Letn linear formsL _i onm variables be given, normalized so that all coefficients have absolute value at most unity. Letw ₁, ...,w _m be real numbers andx ₁, ...,x _m be integers. We sayE _i =L _i (w ₁, ...,w _m )-L _i (x ₁, ...,x _m ) is the error in approximating thew's by thex's with respect to formL _i It is shown that given anyw's there is an integral approximation ofx's so that the errorsE _i are small-roughly that$E_i = O(\sqrt i In i )$ simultaneously for alli.