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

B. Anderson[1]; J. M. Ash[2]; R. L. Jones[3]; D. G. Rider[4]; B. Saffari[5]

• [1] 130 Channing Ln Chapel Hill, NC 27516 (USA)
• [2] DePaul University Department of Mathematical Sciences Chicago, IL 60614 (USA)
• [3] Conserve School 5400 N. Black Oak Lake Drive Land O’Lakes, WI 54540 (USA)
• [4] University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1313 (USA)
• [5] 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 $p=1.$

## How to cite

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Anderson, B., et al. "Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms." Annales de l’institut Fourier 57.5 (2007): 1377-1404. <http://eudml.org/doc/10262>.

@article{Anderson2007,
abstract = {A sum of exponentials of the form $f(x)=\exp \left( 2\pi iN_\{1\}x\right) +\exp \left( 2\pi iN_\{2\}x\right) +\cdots +\exp \left( 2\pi iN_\{m\}x\right)$, 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 $p&gt;1,\,$ and every set $E$ of the torus $\mathbb\{T=R\}/\mathbb\{Z\}$ with $|E| &gt;0,$ there are idempotents concentrated on $E$ in the $L^\{p\}$ sense. More precisely, for each $p&gt;1,$ there is an explicitly calculated constant $C_\{p\}&gt;0$ so that for each $E$ with $|E| &gt;0$ and $\epsilon &gt;0$ one can find an idempotent $f$ such that the ratio $\left( \int _\{E\}|f| ^\{p\}\big / \int _\{\mathbb\{T\}\}|f| ^\{p\} \right) ^\{1/p\}$ is greater than $C_\{p\}-\epsilon$. 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 $L^\{p\}$ concentration phenomenon fails to occur when $p=1.$},
affiliation = {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)},
author = {Anderson, B., Ash, J. M., Jones, R. L., Rider, D. G., Saffari, B.},
journal = {Annales de l’institut Fourier},
keywords = {Idempotents; idempotent trigonometric polynomials; $L^\{p\}$ norms; Dirichlet kernel; concentrating norms; sums of exponentials; $L^\{1\}$ concentration conjecture; weak restricted operators; norms; concentration conjecture},
language = {eng},
number = {5},
pages = {1377-1404},
publisher = {Association des Annales de l’institut Fourier},
title = {Exponential sums with coefficients $0$ or $1$ and concentrated $L^\{p\}$ norms},
url = {http://eudml.org/doc/10262},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Anderson, B.
AU - Ash, J. M.
AU - Jones, R. L.
AU - Rider, D. G.
AU - Saffari, B.
TI - Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1377
EP - 1404
AB - A sum of exponentials of the form $f(x)=\exp \left( 2\pi iN_{1}x\right) +\exp \left( 2\pi iN_{2}x\right) +\cdots +\exp \left( 2\pi iN_{m}x\right)$, 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 $p&gt;1,\,$ and every set $E$ of the torus $\mathbb{T=R}/\mathbb{Z}$ with $|E| &gt;0,$ there are idempotents concentrated on $E$ in the $L^{p}$ sense. More precisely, for each $p&gt;1,$ there is an explicitly calculated constant $C_{p}&gt;0$ so that for each $E$ with $|E| &gt;0$ and $\epsilon &gt;0$ one can find an idempotent $f$ such that the ratio $\left( \int _{E}|f| ^{p}\big / \int _{\mathbb{T}}|f| ^{p} \right) ^{1/p}$ is greater than $C_{p}-\epsilon$. 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 $L^{p}$ concentration phenomenon fails to occur when $p=1.$
LA - eng
KW - Idempotents; idempotent trigonometric polynomials; $L^{p}$ norms; Dirichlet kernel; concentrating norms; sums of exponentials; $L^{1}$ concentration conjecture; weak restricted operators; norms; concentration conjecture
UR - http://eudml.org/doc/10262
ER -

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