### 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) |
---|---|

Pages (from-to) | 679-706 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 289 |

Issue number | 2 |

DOIs | |

State | Published - 1985 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Six standard deviations suffice.** / Spencer, Joel.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 289, no. 2, pp. 679-706. https://doi.org/10.1090/S0002-9947-1985-0784009-0

}

TY - JOUR

T1 - Six standard deviations suffice

AU - Spencer, Joel

PY - 1985

Y1 - 1985

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.

UR - http://www.scopus.com/inward/record.url?scp=84862598391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862598391&partnerID=8YFLogxK

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

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

M3 - Article

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 -