On odd-normal numbers

Abstract

A real number x is considered normal in an integer base \(b \geqslant 2\) if its digit expansion in this base is “equitable”, ensuring that for each \(k \geqslant 1\), every ordered sequence of k digits from \(\{0, 1, \ldots , b-1\}\) occurs in the digit expansion of x with the same limiting frequency. Borel’s classical result [4] asserts that Lebesgue-almost every \(x \in {\mathbb {R}}\) is normal in every base \(b \geqslant 2\). This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension [30] but zero Fourier dimension. The latter condition means that \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) supports a Rajchman measure \(\mu \), whose Fourier transform \({\widehat{\mu }}(\xi )\) approaches 0 as \(|\xi | \rightarrow \infty \) by definiton, albeit slower than any negative power of \(|\xi |\). Moreover, the decay rate of \({\widehat{\mu }}\) is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt [38] and a construction of Lyons [24]. As a consequence, \(\mathscr {N}({\mathscr {O}}, {\mathscr {E}})\) emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem [17] in the special case of \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\).