Reynold Fregoli

Royal Holloway

Bounded Exponential Sums

Ergodic Theory and Dynamical Systems Seminar

5th March 2020, 2:00 pm – 3:00 pm
Fry Building, Fry Building, 2.04

Let $A\subset\mathbb{N}$, $\alpha\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2\pi ix}$. We set $$S_{A}(\alpha,N):=\sum_{\substack{n\in A\\n\leq N}}e(n\alpha).$$ Recently, Lambert A'Campo proposed the following question: is there an infinite non-cofinite set $A\subset\mathbb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this talk I will give an idea of why such sets do not exist. To show this, I use a theorem by Duffin and Schaeffer on complex power series. The above result can also be extended to prove 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, it can be shown that there are infinite non-cofinite sets $A$ such that $|S_{A}(\alpha,N)|$ is bounded for all $\alpha\in E\subset (0,1)$, where $E$ has full Hausdorff dimension and $\mathbb{Q}\cap (0,1)\subset E$.

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