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Six Standard Deviations Suffice

Joel Spencer
Transactions of the American Mathematical Society
Vol. 289, No. 2 (Jun., 1985), pp. 679-706
DOI: 10.2307/2000258
Stable URL: http://www.jstor.org/stable/2000258
Page Count: 28
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Six Standard Deviations Suffice
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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 Kn1/2, 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 pigeonhold 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 p1,⋯,pn ∈ {0,1} may be approximated by ε1,...,εn ∈ {0, 1} so that the forms have small error.

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