A quantitative version of the idempotent theorem in harmonic analysis

Abstract

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ * µ = µ, or alternatively if µ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ G : µ(γ) = 1} belongs to the coset ring of G, that is to say we may writewhere the Γ j are open subgroups of G.In this paper we show that L can be bounded in terms of the norm µ , and in fact one may take L exp exp(C µ 4 ). In particular our result is nontrivial even for finite groups.

Approximate groups. I The torsion-free nilpotent case

“…Furthermore, Proposition 7.2 below together with a little calculation implies that X 2ρ is exp(C s k s )-controlled by X ρ . These two facts imply that the system (X ρ ) ρ 4 forms a Bourgain system in the sense of [9,16], thereby providing a link between our work and that of Sanders. Proof .…”

Chowla’s cosine problem

“…It is more natural to take the measures μ Bρ rather than β ρ ; however, certain positivity requirements in [GS07] precipitated the use of these convolved measures and we shall in fact further leverage this convenience in the proof of Corollary 6.3 below.…”

A Freĭman-type theorem for locally compact abelian groups

“…In a different direction it is natural, as was done in [6], to conjecture Theorem 1.2 in non-abelian groups. …”

“…Multi-dimensional progressions and Bohr sets were both brought under the auspices of so called Bourgain systems in [6].…”

“…Moreover, "approximate groups" are far more abundant than genuine groups, which in many cases makes them more useful. We refer the reader to [2,4,6,8,10] and [12] for examples. It is worth noting that the apparently more general notion of Bourgain system was introduced in [6] as a candidate for "approximate groups", but we shall show in §3 that this notion and ours are essentially equivalent.…”