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1. CJM 2011 (vol 64 pp. 1201)
The Central Limit Theorem for Subsequences in Probabilistic Number Theory Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
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 Categories:60F05, 42A55, 11D04, 05C55, 11K06 |