Weak Gibbs property and system of numeration

On the Gibbs properties of Bernoulli convolutions related to $β$-numeration in multinacci bases

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the $β$-numeration. A matrix decomposition of these measures is obtained in the case when $β$ is a PV number. We also determine their Gibbs properties for $β$ being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

Numbers with integer expansion in the numeration system with negative base

In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-β}$ of numbers whose representation uses only non-negative powers of $-β$, the so-called $(-β)$-integers. We describe the distances between consecutive elements of $\Z_{-β}$. In case that this set is non-trivial we associate to $β$ an infinite word $\boldsymbol{v}_{-β}$ over an (in general infinite) alphabet. The self-similarity of $\Z_{-β}$, i.e., the property $-β\Z_{-β}\subset \Z_{-β}$, allows us to find a morphism under…

非整数基中的Midy定理与斐波那契数的可整除性

Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

将Midy定理推广到非整数基的计数系统，研究分数在基β>1下的展开性质，并刻画了基为黄金比例时满足Midy性质的素数分母q。

Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $μ$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehatμ(ξ)| = O(|ξ|^{-η}), \quad \text{as } |ξ| \to \infty.$$ We show that this property holds for any Gibbs measure $μ$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which exte…