Exponential sums with coefficients 0 or 1 and concentrated L^{p} norms

Abstract

A sum of exponentials of the form f (x) = exp (2πiN 1 x)+exp (2πiN 2 x)+ • • • + exp (2πiN m x), where the N k are distinct integers is called an idempotent trigonometric polynomial (because the convolution of f with itself is f ) or, simply, an idempotent. We show that for every p > 1, and every set E of the torus T = R/Z with |E| > 0, there are idempotents concentrated on E in the L p sense. More precisely, for each p > 1, there is an explicitly calculated constant C p > 0 so that for each E with |E| > 0 and ǫ > 0 one can find an idempotent f such that the ratio E |f | p T |f | p 1/p is greater than C p − ǫ. This is in fact a lower bound result and, though not optimal, it is close to the best that our method gives. We also give both heuristic and computational evidence for the still open problem of whether the L p concentration phenomenon fails to occur when p = 1.titre francais: Sommes d'exponentielles à coefficients 0 ou 1 et concentration de normes L p résumé francais: Une somme d'exponentielles de la formeoù les N k sont des entiers distincts, est appelée un polynôme trigonométrique idempotent (car f * f = f ) ou, simplement, un idempotent. Nous prouvons que pour tout réel p > 1, et tout E ⊂ T = R/Z avec |E| > 0, il existe des idempotents concentrés sur E au sens de la norme * Partially supported by NSF Grant DMS-9707011 † Partially supported by NSF Grant DMS-9531526 L p . Plus précisément, pour tout p > 1, nous calculons explicitement une constante C p > 0 telle que pour tout E avec |E| > 0, et tout réel ǫ > 0, on puisse construire un idempotent f tel que le quotient E |f | p T |f | p 1/p soit supérieur à C p − ǫ.Ceci est en fait un théorème de minoration qui, bien que non optimal, est proche du meilleur résultat que notre méthode puisse fournir. Nous présentons également des considérations heuristiques et aussi numériques concernant le problème (toujours ouvert) de savoir si le phénomène de concentration L p a lieu ou non pour p = 1.mots clefs: idempotents, polynômes trigonométriques idempotents, normes L p , noyau de Dirichlet, concentration de normes, sommes d'exponentielles, conjecture de concentration en norme L 1 , opérateurs faiblement restreints.

Integral concentration of idempotent trigonometric polynomials with gaps

“…This is described recently in the survey [5]. It has since then been the object of considerable interest, with improving lower bounds obtained by Pichorides, Montgomery, Kahane and Ash, Jones and Saffari, see [1,2,3] for details. In 1983 Déchamps-Gondim, Piquard-Lust and Queffélec [13,14] answered a question from [1], proving the precise value for all p > 2.…”

“…The starting point of our work was a conjecture in [3] regarding the impossibility of the concentration of the integral norm of idempotents.…”

“…Before recording the main result of the paper [3], let us give some notations and definitions. We first start by the notion of concentration on symmetric open sets, for which results are more complete, and proofs are more elementary.…”

“…In 1983 Déchamps-Gondim, Piquard-Lust and Queffélec [13,14] answered a question from [1], proving the precise value for all p > 2. As in [13,14,2,3], we will consider the same notion of p-concentration of (convolution-)idempotents for measurable sets, too. Definition 4.…”

“…The proof of [2,3] is based on the properties of the function (7) D n (x)D n (qx), where D n stands for the Dirichlet kernel. We will use the same notation as in [3] and define the Dirichlet kernel as e(νx) = e πi(n−1)x sin(πnx) sin(πx) .…”