Name
Abstract | ||
|---|---|---|
Let G be a finite Abelian group and A a subset of G. The spectrum of A is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime |A|=|G|α whenever α≤c, where c≥1/2 is some absolute constant. On the other hand, there are statistical results, which apply only to a noticeable fraction of the elements, which give nontrivial bounds even to much smaller sets. One such theorem (due to Bourgain) goes as follows. For a noticeable fraction of pairs γ1,γ2 in the spectrum, γ1+γ2 belongs to the spectrum of the same set with a smaller threshold. Here we show that this result can be made combinatorial by restricting to a large subset. That is, we show that for any set A there exists a large subset A′, such that the sumset of the spectrum of A′ has bounded size. Our results apply to sets of size |A|=|G|α for any constant α>0, and even in some sub-constant regime. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1016/j.jcta.2016.11.009 | Journal of Combinatorial Theory, Series A |
Keywords | Field | DocType |
Fourier spectrum,Sum set,Doubling constant | Abelian group,Discrete mathematics,Combinatorics,Sumset,Fourier series,Mathematics,Bounded function | Journal |
Volume | ISSN | Citations |
148 | 0097-3165 | 0 |
PageRank | References | Authors |
0.34 | 1 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| kaave hosseini | 1 | 0 | 1.35 |
| Shachar Lovett | 2 | 520 | 55.02 |