Lp norm local estimates for exponential sums

Abstract

Abstract We prove that for every real number p>1 there is an explicitly calculated constant Kp>0 such that, for any arc J of the one-dimensional torus T=R/Z with |J|>0, no matter how small, one can find some 1-periodic exponential sum f(x)= ∑ m k=1 exp (2iπN k x) so that ( ∫ J |f(x)| p d x) 1/p ≥K p ·( ∫ J |f(x)| p d x) 1/p (Lp norm “local concentration” on J). For p≥2 the result remains true (with another Kp) if the arc J is replaced by any set E⊂ T with |E|>0. The special case p=2 of our results had been studied by many authors about twenty years ago, but the general case p>1 had remained open. The limit case p=1 remains unsettled to this date.