A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime

TL;DR

This paper analyzes the function $f_r(x)=\frac{\log(1+rx)}{\log(1+x)}$ arising from fractal dimensions in number-theoretic constructions. It proves $f_r$ is a complete Bernstein function for $0<r<1$ and a Stieltjes function for $r>1$, providing explicit integral representations with densities $\sigma_r$ and $\omega_r$ and establishing Pick-function boundary behavior via harmonic analysis. The work also derives a convolution equation linking these representations through Laplace transforms and demonstrates that a natural generalization of a known CBF preservation under scaling fails in general. These results connect complex-analytic techniques, potential theory, and probabilistic interpretations, with implications for the study of fractal dimensions and related function classes.

Abstract

Let $f_r(x)=\log(1+rx)/\log(1+x)$ for $x>0$. We prove that $f_r$ is a complete Bernstein function for $0\le r\le 1$ and a Stieltjes function for $1\le r$. This answers a conjecture of David Bradley that $f_r$ is a Bernstein function when $0\le r\le 1$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Remark 5
• Proposition 6
• Corollary 7
• proof
• Remark 8
• Corollary 9