On irregularities of distribution of weighted sums-of-digits
Let \(s(n)\) be the sum-of-digits function of \(n\) in base \(2\). \textit{A. O. Gelfond} [Acta Arith. 13, 259--265 (1968; Zbl 0155.09003)] has shown that \((s(3n))_{n\geq 0}\) is uniformly distributed in the residue classes modulo \(2\). However, \textit{D. J. Newman} [Proc. Am. Math. Soc. 21, 719--721 (1969; Zbl 0194.35004)] was able to prove that \[ \#\{0\leq n<N: s(3n)\equiv 0\bmod 2\}-\frac N2>cN^{\frac{\log 3}{\log 4}}\quad\text{ for all }N. \] This motivated the authors of the paper under review to consider whether also the weighted sum-of-digits function satisfies a Newman-type property. To this end let \(\pmb{\gamma}=(\gamma_0,\gamma_1,\ldots)\in\{0,1\}^{\mathbb{N}_0}\) be a weight sequence. Then the weighted sum-of-digits function \(s_\gamma(n)\) is given by \[ s_\gamma(n)=a_r\gamma_r+\cdots+a_1\gamma_1+a_0\gamma_0 \quad\text{for}\quad n=a_r2^r+\cdots+a_12+a_0. \] Furthermore for a given weight sequence \(\pmb{\gamma}\), and a residue class \(k\in\{0,1,2\}\), we say that \((s_\gamma(3n+k))_{n\geq0}\) satisfies a strong Newman-type property if for some \(c,\kappa>0\) we have \[ \#\{0\leq n<N:s(3n+k)\equiv 0\bmod 2\}-\frac N2>cN^{\kappa} \] or \[ \#\{0\leq n<N:s(3n+k)\equiv 0\bmod 2\}-\frac N2<-cN^{\kappa} \] for all \(N\in\mathbb{N}_0\). Similarly, we say that the sequences \((s_\gamma(3n+k))_{n\geq0}\) satisfies a weak Newman-type property if the above inequalities only hold for sufficiently large \(N\). The authors of the present paper give a complete characterization of weight sequences together with residue classes which satisfy strong, weak or no Newman-type properties. These results have applications in the investigation of the distribution of Niederreiter--Halton sequences.
distribution of Niederreiter--Halton sequences
scientific article; zbMATH DE number 5833544
scientific article