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

TL;DR

This paper demonstrates that for any p > 1, there exist idempotent trigonometric polynomials highly concentrated on any subset of the torus, providing explicit bounds and evidence related to Lp norm concentration.

Contribution

The authors establish the existence of Lp norm concentration for idempotent trigonometric polynomials on arbitrary sets for all p > 1, with explicit constants and bounds.

Findings

Existence of idempotents concentrated on sets of positive measure for p > 1

Explicit constants for concentration bounds are provided

Evidence suggests concentration may fail at p = 1

Abstract

Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f 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 Lp sense. More precisely, for each p > 1, there is an explicitly calculated constant Cp > 0 so that for each E with |E| > 0 and epsilon > 0 one can find an idempotent f such that the pth root of the ratio of the integral over E of the pth power of |f| to the integral over T of the pth power of |f| is greater than Cp - epsilon. 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 Lp concentration phenomenon fails to occur when p = 1.