TY - JOUR

T1 - Six standard deviations suffice

AU - Spencer, Joel

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1985/6

Y1 - 1985/6

N2 - 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(1n 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 p1……, pnÎ (0, 1) may be approximated by e1,&, enÎ (0, 1) so that the forms have small error.

AB - 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(1n 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 p1……, pnÎ (0, 1) may be approximated by e1,&, enÎ (0, 1) so that the forms have small error.

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U2 - 10.1090/S0002-9947-1985-0784009-0

DO - 10.1090/S0002-9947-1985-0784009-0

M3 - Article

AN - SCOPUS:84862598391

VL - 289

SP - 679

EP - 706

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -