On Toric Orbits in the Affine Sieve

TL;DR

This paper analyzes a heuristic model explaining the failure of saturation in the affine sieve with toral Zariski closure, formulates conjectures on classical arithmetic problems, and tests them empirically.

Contribution

It introduces a detailed heuristic model for saturation failure in the affine sieve and proposes precise conjectures supported by empirical data.

Findings

Heuristic model explains saturation failure in specific affine sieve cases.

Formulated conjectures on classical arithmetic problems.

Empirical data supports the proposed conjectures.

Abstract

We give a detailed analysis of a heuristic model for the failure of "saturation" in instances of the Affine Sieve having toral Zariski closure. Based on this model, we formulate precise conjectures on several classical problems of arithmetic interest, and test these against empirical data.