On odd-normal numbers

Abstract

A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, 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 \cite{b09} asserts that Lebesgue-almost every $x \in \mathbb R$ is normal in every base $b \geq 2$. 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 \cite{p81} 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 \cite{s60} and a construction of Lyons \cite{l86}. 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 \cite{Kahane-Salem-64} in the special case of $\mathscr N(\mathscr{O}, \mathscr{E})$.