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.
ASJC Scopus subject areas
- Applied Mathematics