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# The Central Limit Theorem for Subsequences in Probabilistic Number Theory

Published:2011-12-23
Printed: Dec 2012
• Christoph Aistleitner,
Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 8010 Graz, Austria
• Christian Elsholtz,
Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 8010 Graz, Austria
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## Abstract

Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying $$\tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty.$$ If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of $$\tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}$$ converges to a Gaussian distribution. In the case $$1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty$$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$\frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}}$$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.
 Keywords: central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
 MSC Classifications: 60F05 - Central limit and other weak theorems 42A55 - Lacunary series of trigonometric and other functions; Riesz products 11D04 - Linear equations 05C55 - Generalized Ramsey theory [See also 05D10] 11K06 - General theory of distribution modulo $1$ [See also 11J71]

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