Large deviation principles for lacunary sums

Let ( a k stretchy="false">) ? N (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition alttext="a plus 1 slash greater-than q 1"> + 1 / &gt; q encoding="application/x-tex">a_{k+1}/a_k&gt; &gt;1 for all alttext="k encoding="application/x-tex">k\in N , and let S n ? = ?<!-- ? </mml:munderover> cos ?<!-- ? <mml:mn>2 ?<!-- ? <mml:mo>, width="thickmathspace" stretchy="false">[ 0 stretchy="false">] . encoding="application/x-tex">\begin{equation*} S_n(\omega ) = \sum _{k=1}^n \cos (2\pi a_k \omega ), \qquad n\in N, \; \in [0,1]. \end{equation*} Then n"> encoding="application/x-tex">S_n is called lacunary trigonometric sum, can viewed as random variable defined on probability space alttext="normal Omega right-bracket"> mathvariant="normal">?<!-- ? encoding="application/x-tex">\Omega [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that typical independent variables. For example, central limit theorem encoding="application/x-tex">(S_n)_{n\in {N}} has been obtained by Salem Zygmund, while law iterated logarithm due Erd?s Gál. In this paper we study large deviation principles sums. Specifically, under right-arrow normal infinity"> stretchy="false">?<!-- ? mathvariant="normal">?<!-- ? encoding="application/x-tex">a_{k+1}/a_k \to \infty prove encoding="application/x-tex">(S_n/n)_{n does indeed satisfy principle speed alttext="n"> encoding="application/x-tex">n same rate function I overTilde"> I ~<!-- ~ </mml:mover> encoding="application/x-tex">\widetilde {I} variables arcsine distribution. On other hand, show may fail hold when only assume condition. However, in special case Superscript k"> encoding="application/x-tex">a_k= q^k some alttext="q StartSet 3 ellipsis EndSet"> fence="false" stretchy="false">{ 3 …<!-- … stretchy="false">} encoding="application/x-tex">q\in \{2,3,\ldots \} satisfies (with ) q"> encoding="application/x-tex">I_q different from describe algorithm compute arbitrary number terms Taylor expansion . addition, also converges pointwise I encoding="application/x-tex">q\to Furthermore, construct perturbation encoding="application/x-tex">(a_k)_{k encoding="application/x-tex">(2^k)_{k which 2"> 2 encoding="application/x-tex">k\to but at time case, surprisingly encoding="application/x-tex">I_2 one might naïvely expect. We relate fact solutions certain Diophantine equations. Together, these results very sensitive arithmetic This particularly noteworthy since no such effects visible or Our proofs use combination tools theory, harmonic analysis, dynamical systems.