The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory (2007.15936v3)

Abstract: Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of $T_{3}$ being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed $\left(p,q\right)$-adic analysis, the study of functions from the $p$-adics to the $q$-adics, where $p$ and $q$ are distinct primes. In this, the first paper, working with the $T_{q}$ maps as a toy model for the more general theory, for each odd prime $q$, we construct a function $\chi_{q}:\mathbb{Z}{2}\rightarrow\mathbb{Z}{q}$ (the Numen of $T_{q}$) and prove the Correspondence Principle (CP): $x\in\mathbb{Z}\backslash\left{ 0\right}$ is a periodic point of $T_{q}$ if and only there is a $\mathfrak{z}\in\mathbb{Z}{2}\backslash\left{ 0,1,2,\ldots\right}$ so that $\chi{q}\left(\mathfrak{z}\right)=x$. Additionally, if $\mathfrak{z}\in\mathbb{Z}{2}\backslash\mathbb{Q}$ makes $\chi{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$, then the iterates of $\chi_{q}\left(\mathfrak{z}\right)$ under $T_{q}$ tend to $+\infty$ or $-\infty$.

• The paper introduces the numen function as a novel tool to reinterpret Collatz-type maps via non-archimedean analysis.
• It employs (p,q)-adic analysis to examine periodicity and divergence in arithmetic dynamical systems.
• The study establishes a correspondence principle linking periodic points with rational numen values in 2-adic contexts.

Analysis of the Collatz Conjecture and Non-Archimedean Spectral Theory - Part I

Maxwell C. Siegel's paper explores a nuanced exploration of the Collatz Conjecture through the lens of non-archimedean analysis, specifically leveraging (p,q)-adic analysis. Typically elusive and recognized for its mathematical tenacity, the Collatz Conjecture serves as the focal point of this research, which aims to contextualize such problems within a broader analytical paradigm. Siegel introduces a series of innovations, including the concept of the numen—a function associated with Collatz-type maps—which serves to reinterpret classical conjectures into the framework of Non-Archimedean Spectral Theory.

This extensive paper, seemingly the first in a series, sets forth to challenge established arithmetic dynamics approaches, especially in relation to functions mapping the p-adics to the q-adics, where p and q are distinct primes. One prominent element of the analysis is the paper of the Shortened qx + 1 map and formulation through the numen and Correspondence Principle (CP), underscoring points where periodicity and divergence occur in relation to non-archimedean value distributions.

Numerical and Theoretical Insights

The paper proposes a correspondence principle (CP) with practical implications for understanding periodic and divergent behaviors of the Collatz map. Specifically, it establishes that a non-zero periodic point within the map is associated with a rational integer value of the numen across a 2-adic integer input. This groundbreaking finding, though non-constructively proven, broadens insights into the behavior of these maps, suggesting a structured approach to previously inaccessible mathematical enigmas.

Interestingly, the non-archimedean perspective provides rich analytical tools previously overlooked. Siegel’s framework suggests a level of depth in (p,q)-adic analysis, emphasizing the utility and richness of non-continuous functions—those he classifies as rising-continuous—allowing the exploration of new mathematical challenges that may have direct implications for arithmetic dynamics.

Practical and Theoretical Implications

In applying non-archimedean spectral theory, the paper speculates potential advancements not only in characterizing the enigmatic nature of Collatz-type maps but also in propelling forward theories in transcendental number theory and possibly refining tools for evaluating complex dynamical systems. The exploration of divergent points and their impact on the paper of the Collatz map, particularly through the paper of properties like the branch points of the numen, aligns the paper's approaches with large-scale theoretical frameworks previously unexplored in depth with such specificity.

Moving forward, the implications of this work could extend into realms of complex dynamical analyses where other mathematical conjectures remain opaque. It sets a precedent in its innovative method by situating the Collatz Conjecture amidst non-archimedean landscapes and spectral theories, potentially framing new conjectures and morphing our understanding of number theory itself.

Future Directions

The findings and new paradigms introduced invite further scholarly exploration, particularly in the direction of applying non-archimedean spectral frameworks to related areas, including the development of differentiation theories in this context. Additionally, the insights gathered here could significantly impact adjacent fields such as p-adic quantum mechanics. It will be captivating to note how Siegel's continuing series of papers build upon this foundation and the extent to which these concepts will be integrated and expanded within mathematical research.

Siegel’s contribution, situated within a rich analytical universe, fosters an expansion of non-archimedean theories and insights, ideally rejuvenating interest around the notorious Collatz Conjecture and analogous complex problems in mathematical theory and practice.