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ADDITIVE DIMENSION AND A THEOREM OF SANDERS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 100 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 22 October 2015, pp. 124-144
- Print publication:
- February 2016
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A removal lemma for linear configurations in subsets of the circle
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 56 / Issue 3 / October 2013
- Published online by Cambridge University Press:
- 28 June 2013, pp. 657-666
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Freiman Homomorphisms of Random Subsets of
$\mathbb{Z}_{N}$
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- Journal:
- Combinatorics, Probability and Computing / Volume 22 / Issue 4 / July 2013
- Published online by Cambridge University Press:
- 11 June 2013, pp. 592-611
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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.
