Integral approximation sequences

AbstractLetn linear formsLi onm variables be given, normalized so that all coefficients have absolute value at most unity. Letw1, ...,wm be real numbers andx1, ...,xm be integers. We sayEi=Li (w1, ...,wm)-Li (x1, ...,xm) is the error in approximating thew's by thex's with respect to formLi It is shown that given anyw's there is an integral approximation ofx's so that the errorsEi are small-roughly that $$E_i = O(\sqrt i In i )$$ simultaneously for alli.