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.
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References
J. Beck and T. Fiala, “Integer making theorems”,Discrete Applied Mathematics 3 (1981), 1–8.
H.J. Ryser, “Combinatorial mathematics”, Carus Mathematical Monograph 14, Mathematical Association of America (1963).
J. Spencer, “Sequences with small discrepancy relative ton events”,Compositio Mathematica 47 (1982) 365–392.
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Beck, J., Spencer, J. Integral approximation sequences. Mathematical Programming 30, 88–98 (1984). https://doi.org/10.1007/BF02591800
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DOI: https://doi.org/10.1007/BF02591800