### Abstract

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Kn^{1/2}, 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 p_{1}……, p_{n}Î (0, 1) may be approximated by e_{1},&, e_{n}Î (0, 1) so that the forms have small error.

Original language | English (US) |
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Pages (from-to) | 679-706 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 289 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1985 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics