$p$-adic fractal strings of arbitrary rational dimensions and Cantor strings

TL;DR

This paper develops a global theory of complex dimensions for ad extbackslash'elic fractal strings, constructing self-similar $p$-adic and ad extbackslash'elic Cantor strings to explore oscillations and their relation to the Riemann hypothesis.

Contribution

It introduces a natural construction of $p$-adic fractal strings of arbitrary rational dimensions and extends to ad extbackslash'elic Cantor strings as infinite products, advancing the global theory of complex dimensions.

Findings

Constructed self-similar $p$-adic fractal strings of any rational dimension.

Developed ad extbackslash'elic Cantor strings as infinite products of $p$-adic strings.

Proposed a framework linking oscillations of fractal strings to the Riemann hypothesis.

Abstract

The local theory of complex dimensions for real and $p$-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for ad\`elic fractal strings in order to reveal the oscillatory nature of ad\`elic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar $p$-adic fractal strings of any rational dimension in the closed unit interval $[0,1]$. Moreover, as a first step towards a global theory of complex dimensions for ad\`elic fractal strings, we construct an ad\`elic Cantor string in the set of finite ad\`eles $\mathbb{A}_0$ as an infinite Cartesian product of every $p$-adic Cantor string, as well as an ad\`elic Cantor-Smith string in the ring of ad\`eles $\mathbb{A}$ as a Cartesian product of the general Cantor string and the ad\`elic Cantor string.