tailieunhanh - Đề tài " A quantitative version of the idempotent theorem in harmonic analysis "

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, 1. Introduction Let | Annals of Mathematics A quantitative version of the idempotent theorem in harmonic analysis By Ben Green and Tom Sanders Annals of Mathematics 168 2008 1025 1054 A quantitative version of the idempotent theorem in harmonic analysis By Ben Green and Tom Sanders 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 p 2 M G is said to be idempotent if p p p or alternatively if p takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure p is idempotent if and only if the set 7 2 G p 7 1 belongs to the coset ring of G that is to say we may write L p X 1. u. j 1 where the Tj are open subgroups of G. In this paper we show that L can be bounded in terms of the norm p and in fact one may take L 6 expexp C p 4 . In particular our result is nontrivial even for finite groups. 1. Introduction Let us begin by stating the idempotent theorem. Let G be a locally compact abelian group with dual group G. Let M G denote the measure algebra of G that is to say the algebra of bounded regular complex-valued measures on G. We will not dwell on the precise definitions here since our paper will be chiefly concerned with the case G finite in which case M G L1 G . For those parts of our paper concerning groups which are not finite the book 19 may be consulted. A discussion of the basic properties of M G may be found in Appendix E of that book. If p 2 M G satisfies p p p we say that p is idempotent. Equivalently the Fourier-Stieltjes transform p satisfies p2 p and is thus 0 1-valued. The first author is a Clay Research Fellow and is pleased to acknowledge the support of the Clay Mathematics Institute. 1026 BEN GREEN AND TOM SANDERS Theorem Cohen s idempotent theorem . is idempotent if and only if 7 2 G 2 7 1g lies in the coset ring of G that is to say L J X 17 c j 1 where the Tj are open subgroups of G. This result was proved by Paul Cohen 4 . .