Some Fourier series with gaps

Abstract

We examine diverse local and global aspects of the family of Fourier series È n −α e(n k x). In particular, combining number theoretical and harmonic analytic arguments, we study differentiability, Hölder continuity, spectrum of singularities and fractal dimension of the graph.

Multifractal behavior of polynomial Fourier series

“…Proposition 4.2 of [CU07]). It can be that in some cases one can get a stronger result by the methods in [Kat88], but we could not locate a precise statement there.…”

“…In [CJU] is proved that P (x) − P (y) + r is absolutely irreducible over F q when q ∤ deg P and r = 0. Hence by the Riemann hypothesis over finite fields (see Ch.3 of [Sch04]) the corresponding algebraic curve contains q + O q 1/2 points over F q and proceeding as in Proposition 4.4 of [CU07] we have for each q and the O-constant only depends on P (see the details in [CJU]). …”

“…has a long and fascinating history that has generated a vast literature (see [BS86], [CU07,§1], [Dui91] and references). Just to give a glimpse of it in few words, we mention that according to Weierstrass [Wei67], [Edg04], Riemann considered R to be an example of a continuous nowhere differentiable function.…”

“…For k > 2 there is no underlying automorphic function. Actually one can control locally F only in the intervals (a/q − h, a/q + h) with h < q −k whose union is a set of Hausdorff dimension 2/k (see [CU07,§1] and Jarník's theorem [Fal03]) and it has positive measure only for k = 2. A more important barrier to treat the case k > 2 is that using Poisson summation, one improves the trivial estimate only when h < q −k/2 (see the comments in the introduction of §3) while for rational approximations of a point we have to deal with h almost like q −1 (cf.…”

“…The study of the spectrum of singularities of F for P (x) = x k via Poisson summation was initiated in [CU07,§4]. We would like to remark that there is a small typo in that paper: we defined s(x) = lim inf n s n while it should be defined in the same way but restricting n to subsequences n k for which lim k r n k = r(x).…”