Some arithmetic Ramsey problems and inverse theorems

In this dissertation we study arithmetic Ramsey type problems and inverse problems, in various settings. This work consists of two parts. In Part I, we study arithmetic Ramsey type problems over abelian groups. This part consists of three chapters. In Chapter 2, using hypergraph containers, we study the rainbow Erdos-Rothschild problem for sum-free sets. This is joint work with Cheng, Li, Wang, and Zhou. In Chapters 3 and 4, we study the avoidance density for (k,l)-sum-free sets. The upper bound constructions are given in Chapter 3, answering a question asked by Bajnok. We also improved the lower bound for infinitely many (k,l) in both chapters, and a special case of the sum-free conjecture is verified in Chapter 4. These two chapters are based on joint work with Wu. In Part II, we study inverse problems over nonabelian topological groups. Preliminaries to topological groups are given in Chapter 5. In Chapter 6, we first obtain classifications of connected groups and sets which satisfy the equality in Kemperman's inequality, answering a question asked by Kemperman in 1964. When the ambient group is compact, we also get a near equality version of the above result with a sharp exponent bound, which confirms conjectures by Griesmer and by Tao. A measure expansion gap result for simple Lie groups is also presented. This chapter is based on joint work with Tran. In Chapter 7, we study the small measure expansion problem in noncompact locally compact groups. The question that whether there is a Brunn-Minkowski inequality was asked by Henstock and Macbeath in 1953. We obtain such an inequality and prove it is sharp for a large class of groups (including real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc), answering questions by Hrushovski and by Tao. This chapter is based on joint work with Tran and Zhang. This dissertation is based on the following papers and preprints: [41, 108, 107] (Part I), and [105, 106] (Part II).

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Jing, Yifan

Contributors dc:contributor

• Balogh, Jozsef
• Li, Xiaochun
• Ford, Kevin
• van den Dries, Lou
• English, Sean

• arithmetic combinatorics
• Brunn-Minkowski inequality
• inverse problem
• nonabelian groups
• Ramsey problem
• sum-free sets

Statement dc:rights

• Copyright 2021 Yifan Jing

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/112954

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/112954