Six Standard Deviations Suffice

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Kn>/2, K an absolute constant. This improves the basic probabilistic method with which K = c(ln«)1/2. The result is extended to 11 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 Pi_,pn e {0,1} may be approximated by e,.e„ e {0,1} so that the forms have small error. 1. We state our main result first in the language of linear forms. Theorem 1. Let (1.1) L,(xx,...,x„) = a,xxx + ■■■ + ainxn, 1 < i < n, be n linear forms in n variables with all \a i ¡\ < 1. Then there exist £,,...,£„ G {-1, +1} such that (1.2) |L,0,,...,e„)|<7<7n~ for all i, 1 < i < n. Here K is an absolute constant. When all a¡¡ g {0,1} we may consider A = (a¡¡) as the incidence matrix for a family of n sets on n elements. That is, we may set A= {j: ai} =1}. Given a two-coloring, say Red and Blue, of {l,...,n}, the discrepancy disc(A') of a set X cz {l,...,n} is defined as the number of Red points in X minus the number of Blue points in X. If we interpret e, = +1 as meaning /' is to be colored Red and £, = -1 as Blue then we obtain the following result. Corollary 2. Let Ax,...,An cz {l,...,n}. Then there exists a two-coloring of {1,..., n } so that (1.3) |disc(/l,)|<7i7n~ for all i, 1 < i < n. Received by the editors July 1, 1984. 1980 Mathematics Subject Classification. Primary 05B20; Secondary 41A28, 42A61. 1 This work was initiated while the author was an IREX exchange fellow at the Mathematical Institute, Budapest and completed with support of the National Science Foundation. For the many kindnesses and for the creative research environment provided by Institute staff and colleagues—köszönöm szépen] ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 679 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use