Idris Ali Shaik

Quantitative Collatz Descent to Stretched-Logarithmic Scale in Natural Density (Almost all)

This record contains the version submitted to Forum of Mathematics, Sigma on 7 August 2026, together with the corresponding LaTeX source package. The manuscript proves a quantitative stretched-logarithmic descent theorem for the Collatz map in ordinary natural density. For every...

Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

We prove that, in ordinary natural density, almost every positive integer admits a shortcut Collatz iterate of polylogarithmic size within logarithmically many steps, while every iterate up to the same witness remains at most n^(1+beta) for every fixed beta > 0. The result suppli...