Averaging Almost Periodic Functions along Exponential Sequences

Abstract

Abstract. The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form (α n x) n∈N , where α is a fixed real number with |α| > 1 and x ∈ R is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every x ∈ R.

Pure point diffraction

“…Of particular relevance here is the following result (which has been recently improved and generalized to translation bounded measures [4] [5]. Let S be a repetitive and locally finite subset of R d for which DðSÞ is uniformly discrete.…”

“…It is the latter that we are dealing with here. For uniquely ergodic systems, these two notions are the same [4,23]. Let e > 0 and write W ¼ ðW \ ðz I þ W ÞÞ[ ðW n ðz I þ W ÞÞ, splitting the integral accordingly.…”

“…There is a another more common dynamical system associated with S. We declare two locally finite subsets S 0 , S 00 to be close iff, for some large n and some small shift u 2 R d , ðu þ S 0 Þ \ B n ¼ S 00 \ B n . The closure XðSÞ of the R dorbit of a locally finite set S in the resulting topology is called the dynamical hull of S. Provided that DðSÞ is locally finite, XðSÞ is compact and ðXðSÞ; R d Þ is a dynamical system (see [4] for a more general approach in terms of measures).…”

“…In[6], the set P e as defined in(6)is denoted by P e 2 . Making this replacement makes the presentation here easier and makes no difference to the statements of the results 4. A point set S & R d is called repetitive if each cluster, i.e., S \ K for K compact, reappears in S with bounded gaps (resp.…”

Homometric model sets and window covariograms

“…The Z-span of their positions is the so-called Fourier module. Via minimal embedding, it defines an essentially unique cut and project scheme, whose dual in the sense of [16] is a natural setting to describe both model sets in the same cut and project scheme, see [5] for details. In this sense, the restriction in Theorem 1 is not essential.…”