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Freiman Homomorphisms of Random Subsets of $\mathbb{Z}_{N}$

Published online by Cambridge University Press:  11 June 2013

GONZALO FIZ PONTIVEROS
GONZALO FIZ PONTIVEROS*
Affiliation:
IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brazil (e-mail: [email protected])
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Abstract

Given a set $A\subset\mathbb{Z}_{N}$, we say that a function $f\colon A \to \mathbb{Z}_{N}$ is a Freiman homomorphism if f(a)+f(b)=f(c)+f(d) whenever a,b,c,d ∈ A satisfy a+b=c+d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax+b.

Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p=p(N) is A linear with high probability as N → ∞? We show that if p=(2logN − ω(N))1/3N−2/3, where ω(N) → ∞ as N → ∞, then A is not linear with high probability, whereas if p=N−1/2+ε for any ε>0 then A is linear with high probability.

Keywords

Primary 05D40Secondary 11B75
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 592 - 611
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Alon, N. and Spencer, J. (2008) The Probabilistic Method, Wiley Series in Discrete Mathematics and Optimization, Wiley-Interscience.Google Scholar
[2]Babai, L., Simonovits, M. and Spencer, J. (1990) Extremal subgraphs of random graphs. J. Graph Theory 14 599–622.Google Scholar
[3]Balogh, J., Morris, R. and Samotij, W. (2012) Independent sets in hypergraphs. arXiv:1204.6530v1.Google Scholar
[4]Conlon, D. and Gowers, W. T. (2010) Combinatorial theorems in sparse random sets. arXiv:1011.4310v1.Google Scholar
[5]Freiman, G. (1973) Foundations of a Structural Theory of Set Addition, Vol. 3 of Translations of Mathematical Monographs, AMS.Google Scholar
[6]Kim, J. H. and Vu, V. (2000) Concentration of multivariate polynomials and its applications. Combinatorica 20 417–434.Google Scholar
[7]Kohayakawa, Y., Rödl, V. and Łuczak, T. (1996) Arithmetic progressions of length three in subsets of a random set. Acta Arithmetica 75, 133–163.CrossRefGoogle Scholar
[8]Ruzsa, I. (1994) Generalized arithmetical progressions and sumsets. Acta Math. Hungar. 65 379–388.Google Scholar
[9]Saxton, D. and Thomason, A. (2012) Hypergraph containers. arXiv:1204.6595v1.Google Scholar
[10]Schacht, M. (2009) Extremal results for random discrete structures. Submitted.Google Scholar
[11]Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica 27 199–245.Google Scholar
[12]Tao, T. and Vu, V. (2006) Additive Combinatorics, Cambridge University Press.Google Scholar
[13]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 436–452.Google Scholar
[14]Vu, V. (2002) Concentration of non-Lipschitz functions and applications. Random Struct. Alg. 20 262–316.Google Scholar