Arithmetic statistics of algebraic groups

This thesis investigates various problems in the arithmetic statistics of algebraic groups, focusing on the distribution and properties of algebraic groups of number-theoretic interest. New results obtained throughout this thesis are unified by the development and application of methods, primarily from analytic number theory, to count or bound the number of solutions to certain families of multivariate Diophantine equations. This thesis is organised into two main parts. The first part focuses on the arithmetic statistics of integer and rational matrices. More precisely, the topics considered in this part include the statistics of product sets of integer matrices, rational matrices satisfying specified properties and $2\times 2$ integer matrices with a given determinant. The second part is devoted to studying the arithmetic statistics of algebraic subgroups of the $n$-dimensional multiplicative group of rational numbers. Studying the statistics of these subgroups is connected to studying the distributions of solutions to certain classes of multiplicative Diophantine equations. The problems discussed in this part encompass counting solutions to large families of Diophantine equations in growing orthotopes, as well as enumerating multiplicatively dependent integer vectors lying on a given hyperplane.

Grantor dc:publisher

UNSW, Sydney

Year dc:date

2026

Author dc:creator

• Afifurrahman, Muhammad ; https://orcid.org/0000-0002-7400-320X

• arithmetic statistics
• number theory
• algebraic group
• matrix
• anzsrc-for: 490401 Algebra and number theory

Statement dc:rights

• open access
• CC BY 4.0
• free_to_read

Licence

• https://purl.org/coar/access_right/c_abf2
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

en

Identifier

https://doi.org/10.26190/unsworks/32294

OAI identifier oai:identifier

oai:unsworks.library.unsw.edu.au:1959.4/107802