On odd-normal numbers

A real number x is considered normal in an integer base $$b \geqslant 2$$ b ⩾ 2 if its digit expansion in this base is “equitable”, ensuring that for each $$k \geqslant 1$$ k ⩾ 1 , every ordered sequence of k digits from $$\{0, 1, \ldots , b-1\}$$ { 0 , 1 , … , b - 1 } occurs in the digit expansion of x with the same limiting frequency. Borel’s classical result [4] asserts that Lebesgue-almost every $$x \in {\mathbb {R}}$$ x ∈ R is normal in every base $$b \geqslant 2$$ b ⩾ 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 $${\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})$$ N ( O , 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 $${\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})$$ N ( O , E ) cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that $${\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})$$ N ( O , E ) supports a Rajchman measure $$\mu $$ μ , whose Fourier transform $${\widehat{\mu }}(\xi )$$ μ ^ ( ξ ) approaches 0 as $$|\xi | \rightarrow \infty $$ | ξ | → ∞ by definiton, albeit slower than any negative power of $$|\xi |$$ | ξ | . Moreover, the decay rate of $${\widehat{\mu }}$$ μ ^ 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, $$\mathscr {N}({\mathscr {O}}, {\mathscr {E}})$$ N ( O , 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 $${\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})$$ N ( O , E ) .