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Constructing Small Sets that are Uniform in Arithmetic Progressions

Published online by Cambridge University Press:  12 September 2008

A. Razborov
Affiliation:
Steklov Mathematical Institute, Moscow, Russia
E. Szemerédi
Affiliation:
Rutgers University, New Brunswick N. J., USA
A. Wigderson
Affiliation:
Hebrew University, Jerusalem, Israel

Abstract

For every integer N, we explicitly construct a subset of residues mod N of size(log N)o(1) which is nearly uniformly distributed in every arithmetic progression modulo N.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Ajtai, M., Babai, L., Komlós, J., Pudlák, P., Rödl, V., Szeméredi, E. and Turán, G. (1986) Two lower bounds for branching programs. Proc. 18th STOC 3038.Google Scholar
[2]Ajtai, M., Iwaniec, H., Komlós, J., Pintz, J. and Szemerédi, E. (1990) Construction of a thin set with small Fourier coefficients. Bull. London Math. Soc. 22 583590.CrossRefGoogle Scholar
[3]Alon, N. and Roichman, Y. (in press) Random Cayley graphs and expanders. Random Structures and Algorithms.Google Scholar
[4]Beck, J. (1981) Roth's estimate of discrepancy of integer sequences is nearly sharp. Combinatorica 1 319325.Google Scholar
[5]Beck, J. and Chen, W. (1987) Irregularities of Distributions, Cambridge University Press.CrossRefGoogle Scholar
[6]Chung, F. R. K. (1989) Diameters and eigenvalues. J. AMS 2 187196.Google Scholar
[7]Even, G., Goldreich, O., Luby, M., Nisan, N. and Veličković, B. (1992) Approximations of general independent distributions. Proc. 24th STOC 1016.Google Scholar
[8]Galil, Z., Kannan, R. and Szemerédi, E. (1986) On nontrivial separators for k-page graphs, and simulations by nondeterministic one-tape Turing machines. Proc. 18th STOC 3949.Google Scholar
[9]Katz, N. M. (1989) An estimate for character sums. J. AMS 2 197200.Google Scholar
[10]Roth, K. F. (1964) Remark concerning integer sequences. Ada Arith. 9 257260.CrossRefGoogle Scholar
[11]Ruzsa, I. (1987) Essential components. Proc. London Math. Soc. 54 3856.Google Scholar