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

B. Anderson; J. M. Ash; R. L. Jones; D. G. Rider; B. Saffari

•  130 Channing Ln Chapel Hill, NC 27516 (USA)
•  DePaul University Department of Mathematical Sciences Chicago, IL 60614 (USA)
•  Conserve School 5400 N. Black Oak Lake Drive Land O’Lakes, WI 54540 (USA)
•  University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1313 (USA)
•  Université de Paris XI (Orsay) Département de Mathématiques Université de Paris XI (Orsay) 91405 Orsay Cedex (France)
• Volume: 57, Issue: 5, page 1377-1404
• ISSN: 0373-0956

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## Abstract

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A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, 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 $p>1,\phantom{\rule{0.166667em}{0ex}}$ 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 $ϵ>0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_{E}{|f|}^{p}/{\int }_{𝕋}{|f|}^{p}\right)}^{1/p}$ is greater than ${C}_{p}-ϵ$. 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 $p=1.$

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