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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Six standard deviations suffice


Author: Joel Spencer
Journal: Trans. Amer. Math. Soc. 289 (1985), 679-706
MSC: Primary 05A05
MathSciNet review: 784009
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Abstract: Given $ n$ sets on $ n$ elements it is shown that there exists a two-coloring such that all sets have discrepancy at most $ K{n^{1/2}}$, $ K$ an absolute constant. This improves the basic probabilistic method with which $ K = c{(\ln n)^{1/2}}$. The result is extended to $ n$ finite sets of arbitrary size. Probabilistic techniques are melded with the pigeonhole principle. An alternate proof of the existence of Rudin-Shapiro functions is given, showing that they are exponential in number. Given $ n$ linear forms in $ n$ variables with all coefficients in $ [ - 1, + 1]$ it is shown that initial values $ {p_1}, \ldots ,{p_n} \in \{ 0,1\} $ may be approximated by $ {\varepsilon _1}, \ldots ,{\varepsilon _n} \in \{ 0,1\} $ so that the forms have small error.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0784009-0
PII: S 0002-9947(1985)0784009-0
Article copyright: © Copyright 1985 American Mathematical Society