**A Note on Bounded Exponential Sums.** / Fregoli, Reynold.

Research output: Contribution to journal › Article › peer-review

E-pub ahead of print

**A Note on Bounded Exponential Sums.** / Fregoli, Reynold.

Research output: Contribution to journal › Article › peer-review

Fregoli, R 2020, 'A Note on Bounded Exponential Sums', *Bulletin of the London Mathematical Society*. https://doi.org/10.1112/blms.12431

Fregoli, R. (2020). A Note on Bounded Exponential Sums. *Bulletin of the London Mathematical Society*. https://doi.org/10.1112/blms.12431

Fregoli R. A Note on Bounded Exponential Sums. Bulletin of the London Mathematical Society. 2020 Nov 11. https://doi.org/10.1112/blms.12431

@article{f465591ebc754ac0a850d988854f1e26,

title = "A Note on Bounded Exponential Sums",

abstract = "Let $A\subset\mb{N}$, $\alpha\in(0,1)$, and $e(x):=e^{2\pi ix}$ for $x\in\mb{R}$. We set$$S_{A}(\alpha,N):=\sum_{\substack{n\in A\\n\leq N}}e(n\alpha).$$Recently, Lambert A'Campo posed the following question: is there an infinite non-cofinite set $A\subset\mb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(\alpha,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A\subset\mb{N}$ such that $|S_{A}(\alpha,N)|$ is bounded independently of $N$ for all $\alpha\in E\subset (0,1)$, where $\mb{Q}\cap (0,1)\subset E$ and $E$ has full Hausdorff dimension.",

keywords = "Exponential Sums, Duffin and Schaeffer",

author = "Reynold Fregoli",

year = "2020",

month = nov,

day = "11",

doi = "10.1112/blms.12431",

language = "English",

journal = "Bulletin of the London Mathematical Society",

issn = "0024-6093",

publisher = "Oxford University Press",

}

TY - JOUR

T1 - A Note on Bounded Exponential Sums

AU - Fregoli, Reynold

PY - 2020/11/11

Y1 - 2020/11/11

N2 - Let $A\subset\mb{N}$, $\alpha\in(0,1)$, and $e(x):=e^{2\pi ix}$ for $x\in\mb{R}$. We set$$S_{A}(\alpha,N):=\sum_{\substack{n\in A\\n\leq N}}e(n\alpha).$$Recently, Lambert A'Campo posed the following question: is there an infinite non-cofinite set $A\subset\mb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(\alpha,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A\subset\mb{N}$ such that $|S_{A}(\alpha,N)|$ is bounded independently of $N$ for all $\alpha\in E\subset (0,1)$, where $\mb{Q}\cap (0,1)\subset E$ and $E$ has full Hausdorff dimension.

AB - Let $A\subset\mb{N}$, $\alpha\in(0,1)$, and $e(x):=e^{2\pi ix}$ for $x\in\mb{R}$. We set$$S_{A}(\alpha,N):=\sum_{\substack{n\in A\\n\leq N}}e(n\alpha).$$Recently, Lambert A'Campo posed the following question: is there an infinite non-cofinite set $A\subset\mb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(\alpha,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A\subset\mb{N}$ such that $|S_{A}(\alpha,N)|$ is bounded independently of $N$ for all $\alpha\in E\subset (0,1)$, where $\mb{Q}\cap (0,1)\subset E$ and $E$ has full Hausdorff dimension.

KW - Exponential Sums

KW - Duffin and Schaeffer

U2 - 10.1112/blms.12431

DO - 10.1112/blms.12431

M3 - Article

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

ER -