Bounds in Cohen’s Idempotent Theorem

Abstract

Suppose that G is a finite Abelian group and write WpGq for the set of cosets of subgroups of G. We show that if f : G Ñ Z has }f } ApGq ď M then there is somezpW q1 W and }z} ℓ1pWpGqq " exppM 4`op1q q.

Boolean functions with small spectral norm, revisited

“…One could also proceed using Chang's Lemma [TV06, Lemma 4.35]. We have chosen the former approach because it is closer to the argument in [San16,§7], where the appropriate localisation of Chang's Lemma would be more involved.…”

Lower bounds for the Wiener norm in $\mathbb{Z}_p^d$

“…Besides, it was shown in the work [KS1] that the results of Sanders [Sand2] imply bounds for dense and near-dense subsets.…”

“…In the work of Green and Konyagin [GK] and in several subsequent papers [Sand1], [KS1], [KS2], [Sch], [Sand2] a discrete analog of the Littlewood conjecture (for the case of group Z p ) has been studied. We need some basic definitions (see, for example, [TV], Chapter 4).…”

“…Theorem D [Sand2]. Let G be a finite abelian group and denote by W(G) the set of cosets of subgroups of G. Then for any function f : G → Z with f 1 K there exists some z : W(G) → Z such that…”

“…It is convenient for us to formulate Theorem 3 in such a "conditional" form, in order to not depend on the best known results in one-dimensional case. Moreover, as was mentioned before, for various classes of functions various bounds are known: for instance, in [Sand2] integer-valued functions are studied, whereas in [KS1] and [KS2] indicators functions are learned, and this paper we consider functions which take values in the set {z ∈ C : |z| 1}.…”