2025/06/03 by Joe Clanin, Clanin, Joe, Matthew Rayman +1 · 1 citation
Computer Science · Mathematics · #03D32 #11K16 #68P30 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Information Theory (cs.IT) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2506.02332
openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
A 1952 result of Davenport and Erdős states that if p is an integer-valued polynomial, then the real number 0.p(1)p(2)p(3)… is Borel normal in base ten. A later result of Nakai and Shiokawa extends this result to polynomials with arbitrary real coefficients and all bases b≥ 2. It is well-known that finite-state dimension, a finite-state effectivization of the classical Hausdorff dimension, characterizes the Borel normal sequences as precisely those sequences of finite-state dimension 1. For an infinite set of natural numbers, and a base b≥ 2, the base b Copeland-Erdős sequence of A, CEb(A), is the infinite sequence obtained by concatenating the base b expressions of the numbers in A in increasing order. In this work we investigate the possible relationships between the finite-state dimensions of CEb(A) and CEb(p(A)) where p is a polynomial. We show that, if the polynomial is permitted to have arbitrary real coefficients, then for any s,s^′ in the unit interval, there is a set A of natural numbers and a linear polynomial p so that the finite-state dimensions of CEb(A) and CEb(p(A)) are s and s^′ respectively. The corresponding result for strong finite-state dimension is also shown. We demonstrate that linear polynomials with rational coefficients do not change the finite-state dimension of any Copeland-Erdős sequence, but there exist polynomials with rational coefficients of every larger integer degree that change the finite-state dimension of some sequence. We also prove the surprising fact that there exist sets A and integer-valued monomials p such that CEb(A) is normal, but CEb(p(A)) has finite-state dimension strictly less than one.