HARDY-LITTLEWOOD SERIES AND EVEN CONTINUED FRACTIONS TANGUY RIVOAL AND STÉPHANE SEURET

Abstract. For any s ∈ (1/2, 1], the series Fs(x) = n=1 e ipin2x/ns converges almost everywhere on [−1, 1] by a result of Hardy-Littlewood concerning the growth of the sums∑N n=1 e ipin2x, but not everywhere. However, there does not yet exist an intrinsic descrip-tion of the set of convergence for Fs. In this paper, we define in terms of even continued fractions a subset of points of [−1, 1] of full measure where the series converges. As an intermediate step, we prove that, for s> 0, the sequence of functions N∑ n=1 eipi