Some arithmetic Ramsey problems and inverse theorems

Jing, Yifan

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https://hdl.handle.net/2142/112954 Copy

Description

Title

Some arithmetic Ramsey problems and inverse theorems

Author(s)

Jing, Yifan

Issue Date

2021-06-17

Director of Research (if dissertation) or Advisor (if thesis)

• Balogh, Jozsef
• Li, Xiaochun

Doctoral Committee Chair(s)

Ford, Kevin

Committee Member(s)

• van den Dries, Lou
• English, Sean

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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).

Graduation Semester

2021-08

Type of Resource

Thesis

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

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Copyright 2021 Yifan Jing

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