In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud's monograph Distribution modulo one and Diophantine approximation, which goes back to Mend\`es France. The problem states that the spectrum of the sequence $(\xi\alpha^n)_{n\ge1}$, that is, the set of irrational $\theta\in(0,1)$ for which $(\xi\alpha^n-n\theta)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi\ne0$ and $\alpha>1$. More precisely, we prove that for every real $\alpha>1$ there are $2^{\aleph_0}$ real numbers $\xi>0$ for which the spectrum of $(\xi\alpha^n)_{n\ge1}$ contains one and the same uncountable set.
For a real transcendental number $\xi$, let $\omega_n^*(\xi)$ denote the supremum of all $\omega$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $\leq n$ satisfying $|\xi-\alpha|\leq H(\alpha)^{-\omega -1}$, where $H(\alpha)$ is the naive height of the minimal polynomial of $\alpha$. A...
Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n...
T. Minamide, Haruka Sakai, Y. Tanigawa· 1 citation
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that \[ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. \] By his variation formula,...
V. Botta, K. Castillo, L. Tertuliano da Silva· 0 citations
We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the ou...
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic...
Adam Kanigowski, Kaitlyn Loyd· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
