Problems of almost everywhere convergence related to harmonic analysis and number theory

Abstract

The object of this paper is to discuss certain methods for studying almost everywhere convergence problems. We consider the generalization of the Riesz-Raikov theorem where the dilation numberϑ>1 is not necessarily an integer. It is known (see [B2]) that the averages (1/N)Σ ^N₁ f(ϑ ⁿ x) converge a.e. to ∝ ¹₀ fdx wheneverϑ is algebraic andf a 1-periodic function onR satisfying ∝ ¹₀ |f(x)|² dx<∞. Here the particular case of rational dilation is treated. The reader is referred to [B2] for the general (algebraic) case.

The following definitive relation between a.e. convergence and algebraic numbers is proved. Let {μ _j} be the sequence of measures$\mu _j = \mathop * \limits_{k = 1}^j \left( {\frac{1}{2}\delta _{ - \xi } + \frac{1}{2}\delta _\xi k} \right), \xi = \theta ^{ - 1}$ converging weak* to the natural measureμ on the Cantor set of dissection ratioϑ. Thenf*μ _j→f*μ a.e. for allL ^∞ (T) functions iffϑ is algebraic. This fact depends on [B3] and a variant of Rota’s theorem [Ro] on a.e. convergence of certain compositions of operators. Further applications of this result in ergodic theory are presented in the last section of the paper. In section 4, a.e. convergence of Riemann sums of periodicL ²-functions is investigated. It is shown that almost surelyR _n f has a logarithmic density, where$R_n f\left( x \right) = \frac{1}{n}\sum\limits_0^{n - 1} {f\left( {x + \frac{j}{n}} \right)}$. This result complements the work of R. Salem on the subject.