Asymptotic growth of translation-dilation orbits

TL;DR

This paper proves Manin's conjecture for certain algebraic varieties related to the translation-dilation group over the rationals, using advanced Dirichlet series techniques, though some secondary terms are still unresolved.

Contribution

It completes the proof of Manin's conjecture for specific compactifications of the translation-dilation group, advancing understanding of rational points distribution.

Findings

Confirmed Manin's conjecture for the class of varieties studied.

Developed new methods involving Clausen-like multiple Dirichlet series.

Identified limitations in capturing secondary terms.

Abstract

By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.