Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/0508001

Dimensions of Copeland-Erdos Sequences

Xiaoyang Gu·Jack H. Lutz·Philippe Moser
View original source ↗Read source document ↗Export metadata ↓

Abstract

The base-$k$ {\em Copeland-Erdös sequence} given by an infinite set $A$ of positive integers is the infinite sequence $\CE_k(A)$ formed by concatenating the base-$k$ representations of the elements of $A$ in numerical order. This paper concerns the following four quantities. The {\em finite-state dimension} $\dimfs (\CE_k(A))$, a finite-state version of classical Hausdorff dimension introduced in 2001. The {\em finite-state strong dimension} $\Dimfs(\CE_k(A))$, a finite-state version of classical packing dimension introduced in 2004. This is a dual of $\dimfs(\CE_k(A))$ satisfying $\Dimfs(\CE_k(A))$ $\geq \dimfs(\CE_k(A))$. The {\em zeta-dimension} $\Dimzeta(A)$, a kind of discrete fractal dimension discovered many times over the past few decades. The {\em lower zeta-dimension} $\dimzeta(A)$, a dual of $\Dimzeta(A)$ satisfying $\dimzeta(A)\leq \Dimzeta(A)$. We prove the following. $\dimfs(\CE_k(A))\geq \dimzeta(A)$. This extends the 1946 proof by Copeland and Erdös that the sequence $\CE_k(\mathrm{PRIMES})$ is Borel normal. $\Dimfs(\CE_k(A))\geq \Dimzeta(A)$. These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in $[0,1]$ satisfying the four above-mentioned inequalities.

Explore related subjects

cs.CCcs.ITmath.IT

Keep this discovery

BibTeXRIS

Xiaoyang Gu, Jack H. Lutz, Philippe Moser. 2005-07-30. Dimensions of Copeland-Erdos Sequences. https://arxiv.org/abs/cs/0508001

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗