Related papers: Cantor series constructions of sets of normal numb…

We define and study distributions in R^{d} that we call q-Normal. For q=1 they are really multidimensional Normal, for q\in(-1,1) they have densities, compact support and many properties that resemble properties of ordinary multidimensional…

Let a be a positive integer which is not a perfect h-th power with h greater than 1, and Q_a(x;4,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo 4. When j=0, 2, it is known that…

For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ for which there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q…

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial…

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first…

In this paper we study the distribution of the real algebraic numbers. Given an interval $I$, a positive integer $n$ and $Q>1$, define the counting function $\Phi_n(Q;I)$ to be the number of algebraic numbers in $I$ of degree $n$ and height…

Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely…

For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Let $s_n(x)=\sum_{j=1}^n a_j(x)$. The Hausdorff dimensions of the level sets $E_{\varphi(n),\alpha}:=\{x\in(0,1):…

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference…

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower…

Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random…

Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i…

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. We show that with the additional factor of log q.loglog q the statement is false.…

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as…

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are…

Assume that $m,s\in\mathbb N$, $m>1$, while $f$ is a polynomial with integer coefficients, $\text{deg}~f>1$, $f^{(i)}$ is the $i$th iteration of the polynomial $f$, $\kappa_n$ has a discrete uniform distribution on the set $\{0,1,\ldots,m^n…

In this work, we prove the joint convergence in distribution of $q$ variables modulo one obtained as partial sums of a sequence of i.i.d. square integrable random variables multiplied by a common factor given by some function of an…

For a holomorphic function f on a complex manifold M we explain in this article that the distribution associated to |f | 2$\alpha$ (Log|f | 2) q f --N by taking the corresponding limit on the sets {|f | $\ge$ $\epsilon$} when $\epsilon$…

We show that there exist $(d-1)$ - Ahlfors regular compact sets $E \subset \mathbb{R}^{d}, d\geq 2$ such that for any $t< d-1$, we have \[ \sup_T \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^t}<\infty \] where the supremum is over all tubes $T$…