Idris Ali Shaik

Gallai's conjecture with two exceptional even vertices

Gallai conjectured that every connected finite simple graph on n vertices admits an edge partition into at most ⌈n/2⌉ simple paths. Botler and Sambinelli proved this when every even-degree vertex has at most three even neighbours. Fan, Hou and Zhou exempted one designated even ve...

Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

Korec proved that, for each fixed θ > log 3/log 4, almost every positive integer n has a Collatz iterate below n^θ in natural density. Tao proved descent below every prescribed diverging threshold in logarithmic density. We obtain polylogarithmic descent in ordinary natural densi...

Quantitative Endpoint Transport for the Collatz Map with a Natural-Density Descent Application

Version 2.2.0 isolates and strengthens a reusable arbitrary-target endpoint-transport theorem for the shortcut Collatz map. A power-saving exceptional-set estimate pulls back through one floor(log_2 n)-step endpoint with exponent asymptotically linear in the input density exponen...

Quantitative Endpoint Transport for the Collatz Map with a Natural-Density Descent Application

Version 2.2.0 isolates and strengthens a reusable arbitrary-target endpoint-transport theorem for the shortcut Collatz map. A power-saving exceptional-set estimate pulls back through one floor(log_2 n)-step endpoint with exponent asymptotically linear in the input density exponen...

Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs

Let GX be the graph on the positive integers n ≤ X in which n is joined to np whenever p is prime and np ≤ X. Let AX be its adjacency matrix and let π(X) count the primes up to X. The paper proves λ1(AX)2 = π(X) + 4 ∑p ≤ X 1/(p−1) + o(1). More generally, after deleting a fixed fi...

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...