Real Numbers with Bounded Digit Averages

Abstract

This paper considers numeration schemes, defined in terms of dynamical systems and studies the set of reals which obey some constraints on their digits. In this general setting, (almost) all sets have zero Lebesgue measure, even though the nature of the constraints and the numeration schemes are very different. Sets of zero measure appear in many areas of science, and Hausdorff dimension has shown to be an appropriate tool for studying their nature. Classically, the studied constraints involve each digit in an independent way. Here, more global conditions are studied, which only provide constraints on each digit prefix. The main example of interest deals with reals whose all the digit prefix averages in their continued fraction expansion are bounded by M. More generally, a weight function is defined on the digits, and the weighted average of each prefix has to be bounded by M. This setting can be translated in terms of random walks where each step performed depends on the present digit, and walks under study are constrained to be always under a line of slope M. We first provide a characterization of the Hausdorff dimension 5M, in terms of the dominant eigenvalue of the weighted operator transfer relative to the dynamical system, in a quite general setting. We then come back to our main example; With the previous characterization at hand and use of the Mellin transform, we exhibit the behaviour of $\left| {S_M - 1} \right|$ when the bound M becomes large. Even if this study seems closely related to previous works in Multifractal Analysis, it is in a sense complementary, because it uses weights on digits which grow faster and deals with different methods. This paper only presents a detailed abstract of this study; it describes the main tools and the sketches of proofs whereas a full paper [6] contains all the proofs of the statements which are briefly described here.