相关论文: Borel complexity of the set of typical numbers

We show that the set of absolutely normal numbers is $\mathbf \Pi^0_3$-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is $\Pi^0_3$-complete in the effective Borel hierarchy.

We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various…

We show that normality for continued fractions expansions and normality for base-$b$ expansions are maximally logically separate. In particular, the set of numbers that are normal with respect to the continued fraction expansion but not…

Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $\boldsymbol{\Pi}^0_3$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel…

In this paper, we consider recurrence sequences $x_n=\xi_1 \alpha_1^n+\xi_2 \alpha_2^n$ ($n=0,1,\ldots$) with companion polynomial $P(X)$. For example, the sequence $x_n=\xi_1(4+\sqrt{2})^n+\xi_2(4-\sqrt{2})^n$ satisfies the recurrence…

We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by…

After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic…

Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider…

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of…

Let $X$ be an uncountable Polish space and let $\mathcal{I}$ be an ideal on $\omega$. A point $\eta \in X$ is an $\mathcal{I}$-limit point of a sequence $(x_n)$ taking values in $X$ if there exists a subsequence $(x_{k_n})$ convergent to…

We show that it is decidable whether a given a regular tree language belongs to the class ${\bf \Delta^0_2}$ of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.

We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that,…

In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to…

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes $\mathcal D_2 (\mathbf\Pi^0_3)$ and $\mathcal D_\omega (\mathbf \Pi^0_3)$, that is, the class of sets…

We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a…

Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and…

Let $\mathscr{N}(b)$ be the set of real numbers which are normal to base $b$. A well-known result of H. Ki and T. Linton is that $\mathscr{N}(b)$ is $\boldsymbol{\Pi}^0_3$-complete. We show that the set $\mathscr{N}(b)$ of reals which…

A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…

A real number is called simply normal to base $b$ if every digit $0,1,\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. A real number is called normal to base $b$ if it is simply normal to every base $b,…

This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of…