Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${aⁿ/n}_{n=1}^{\infty }$ is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length $c{N}^{-0.475}$ contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

Soit $D^{*}\left(N\right)$ la discrépance “à l’origine” de la suite $\left(\left\{n\frac{1+\sqrt{5}}{2}\right\}\right)$. Nous montrons que $lim\; sup\frac{D^{*}\left(N\right)}{\mathrm{Log}N}=\frac{3}{20}{\left(\mathrm{Log}\frac{1+\sqrt{5}}{2}\right)}^{-1}=0.31\cdots$, quantité inférieure à celle correspondant à la suite de van der Corput. Les techniques utilisées sont celles liées au développement en fraction continue.

On étudie la discrépance absolue de la suite de Farey d’ordre $n$ et on montre, en utilisant notamment une majoration d’une intégrale portant sur la fonction sommatoire de la fonction de Möbius, qu’elle est égale à $\frac{1}{n}$ exactement, ce qui est la valeur locale au point d’abscisse $\frac{1}{n}$.

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ${\left(\left[n\alpha \right]\beta \right)}_{n\ge 0}$, is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show...