Diophantine equations and the LIL for the discrepancy of sublacunary sequences (Q610622)

The author gives simple and nearly optimal sufficient conditions for the exact law of the iterated logarithm (LIL) for the discrepancy of \((n_k x)\) for a large class of sublacunary growing sequences \((n_k)\) of integers. The value \(1/2\) of the lim sup is exactly the same as in the Chung-Smirnov LIL for i.i.d. random variables. The proof uses methods similar to those of \textit{I. Berkes, W.~Philipp} and \textit{R. F. Tichy} [Ill. J. Math. 50, No. 1-4, 107--145 (2006; Zbl 1145.11058)]. The admissible sequences \((n_k)\) are characterized by a density condition \((\mathbf{K}_\alpha)\) corresponding to a Kolmogorov type condition for the random variables \(X_k = \sum_{2^k \leq j \leq 2^{k+1}} f(n_j x)\), \(k = 1, 2, \dots\) (where \(f\) is a \(1\)-periodic function with bounded variation integrating to zero over \([0,1]\)) and two two-term Diophantine conditions capturing the number-theoretic properties of \((n_k)\). They impose a bound on the number of solutions of the Diophantine equations \(j_1 n_{k_1} - j_2 n_{k_2} = b\), \(b \neq 0\) (condition \((\mathbf{D}_\delta)\), ensuring that the fluctuation behavior of the system is not too wild) and \(j_1 n_{k_1} - j_2 n_{k_2} = 0\) (condition \((\mathbf{D}_\gamma^0)\), controlling the asymptotic variance \(\sigma_N^2 := \int_0^1 ( \sum_{k = 1}^N f( n_k x ) )^2 d x\)). The author shows that from the two conditions \(( \mathbf{K}_\alpha )\), \((\mathbf{D}_\delta)\) and the requirement that \(\sigma_N^2\) grows at least linearly in \(N\) it follows that \(\sum_{k = 1}^N f(n_k x)\) can be approximated by a Wiener process (which in turn implies the central limit theorem and LIL for \(\sum_{k = 1}^N f(n_k x)\)). Also required in the assumptions is that the density and the Diophantine behavior of \(( n_k )\) are connected (expressed as \(\alpha + \delta < 1\), where \(\alpha\), \(\delta\) [and \(\gamma\)] are exponents of \(N\)) in the sense that for a dense sequence \((n_k)\) the Diophantine condition has to be strong and vice versa. The condition \((\mathbf{D}_\gamma^0)\) is then needed to arrive at the exact LIL for the discrepancy of \((n_k x)\).