Estimates from above of certain double trigonometric sums
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In this paper we consider double trigonometric sums. Expressions of this type appear in some problems of quantum chaos and number theory. We are interested in rotation numbers of bounded type. We prove a uniform linear bound on double trigonometric sums along the subsequence of denominators of the continued fraction. The proof uses elementary techniques and the analysis of cancellations in sums of certain oscillatory functions over rotations. We also include a proof of a result on discrepancy for rotations of bounded type and in the Appendix we give an elementary proof of a result by Hardy and Littlewood.