Six standard deviations suffice
HTML articles powered by AMS MathViewer
- by Joel Spencer PDF
- Trans. Amer. Math. Soc. 289 (1985), 679-706 Request permission
Abstract:
Given $n$ sets on $n$ elements it is shown that there exists a two-coloring such that all sets have discrepancy at most $K{n^{1/2}}$, $K$ an absolute constant. This improves the basic probabilistic method with which $K = c{(\ln n)^{1/2}}$. 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 ${p_1}, \ldots ,{p_n} \in \{ 0,1\}$ may be approximated by ${\varepsilon _1}, \ldots ,{\varepsilon _n} \in \{ 0,1\}$ so that the forms have small error.References
- József Beck and Tibor Fiala, “Integer-making” theorems, Discrete Appl. Math. 3 (1981), no. 1, 1–8. MR 604260, DOI 10.1016/0166-218X(81)90022-6
- József Beck and Joel Spencer, Integral approximation sequences, Math. Programming 30 (1984), no. 1, 88–98. MR 755116, DOI 10.1007/BF02591800
- Daniel J. Kleitman, On a combinatorial conjecture of Erdős, J. Combinatorial Theory 1 (1966), 209–214. MR 200179
- John E. Olson and Joel H. Spencer, Balancing families of sets, J. Combin. Theory Ser. A 25 (1978), no. 1, 29–37. MR 480052, DOI 10.1016/0097-3165(78)90028-6
- Walter Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855–859. MR 116184, DOI 10.1090/S0002-9939-1959-0116184-5
- Joel Spencer, Sequences with small discrepancy relative to $n$ events, Compositio Math. 47 (1982), no. 3, 365–392. MR 681615
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 289 (1985), 679-706
- MSC: Primary 05A05
- DOI: https://doi.org/10.1090/S0002-9947-1985-0784009-0
- MathSciNet review: 784009