Filtered rays over iterated absolute differences on layers of integers

TL;DR

This paper explores the complex behavior of iterated high-order gaps in sequences of natural numbers, extending the analysis to a hexagonal and helicoidal geometric structure, and identifies classes of sequences with regularity properties.

Contribution

It introduces a novel geometric extension of the iterated gap system, revealing conditions under which the helicoidal levels remain consistent with the base level, and analyzes the distribution properties of these sequences.

Findings

Existence of sequences with helicoids matching base levels.

Uniform distribution of zero and non-zero elements in certain sequences.

The geometric structure extends to an infinite helicoidal surface.

Abstract

The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth-Glibreath Conjecture for prime numbers. We introduce a natural extension of the original triangular arrangement, obtaining a growing hexagonal covering of the plane. This is just the base level of what further becomes an endless discrete helicoidal surface. % Although the repeated calculation of higher-order gaps causes the numbers that generate the helicoidal surface to decrease, there is no guarantee, and most often it does not even happen, that the levels of the helicoid have any regularity, at least at the bottom levels. However, we prove that there exists a large and nontrivial class of sequences with the property that their helicoids have all levels coinciding with their base levels. This class includes in particular many ultimately binary sequences with a special header. % For almost all of these sequences, we additionally show that although the patterns generated by them seem to fall somewhere between ordered and disordered, exhibiting fractal-like and random qualities at the same time, the distribution of zero and non-zero numbers at the base level has uniformity characteristics. Thus, we prove that a multitude of straight lines that traverse the patterns encounter zero and non-zero numbers in almost equal proportions.