Three problems in additive combinatorics
<p>The problems considered in this thesis involve several fundamental objects in additive combinatorics. In this project, we mainly focus on the structure of subsets of integers. As integers are the most fundamental objects in number theory, one of our main interests in this project is to unde...
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| Format: | Thesis |
| Language: | English |
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2021
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| _version_ | 1826307937216233472 |
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| author | Wang, R |
| author2 | Sanders, T |
| author_facet | Sanders, T Wang, R |
| author_sort | Wang, R |
| collection | OXFORD |
| description | <p>The problems considered in this thesis involve several fundamental objects in additive combinatorics. In this project, we mainly focus on the structure of subsets of integers. As integers are the most fundamental objects in number theory, one of our main interests in this project is to understand additive combinatorial problems related to sets which arise naturally in number theory, such as prime numbers and smooth numbers.</p>
<p>In Chapter 1, we study the size of subsets of smooth numbers which are free of geometric progressions. By taking a Freiman isomorphism, we shall use additive results to study the multiplicative structure in smooth numbers. Our result is proved in Section 1.2 and Section 1.3. For completeness, we illustrate a typical counterexample in Section 1.4, namely the set of <em>k</em>-free smooth numbers. </p>
<p>In the second problem, we focus on difference sets which do not contain shifted primes. Let ℙ denote the set of the prime numbers. Suppose <em>A</em> ⊆ [<em>N</em>] is nonempty and it satisfies (<em>A-A</em>) ∩ (ℙ - 1) = ∅. We are primarily interested in estimating the cardinality of the set <em>A</em> In Chapter 2, we prove a new upper bound on the cardinality of <em>A</em>. In Chapter 3, we investigate sets <em>A</em> such that <em>nA</em> - <em>nA</em> avoids ℙ - 1. We first prove an upper bound on |<em>A</em>| which depends on <em>n</em>, then we generalise an example of Ruzsa.</p>
<p>In the third problem, we are interested in the quantitative aspect of partition regularity of the equation <em>x</em>+<em>y</em> = <em>z</em> when <em>x,y,z</em> are restricted to shifted primes ℙ - 1. In Chapter 4, we provide an exposition of the work of Li and Pan, which is the first result concerning partition regularity over shifted primes. We focus on the analytical ingredients of their transference principle and the quantitative bound that follows from it. In Chapter 5, we improve the bound produced by the method of Li and Pan using a different approach. The main ingredient of the proof is an iteration lemma from Chapter 2.</p>
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| first_indexed | 2024-03-07T07:10:40Z |
| format | Thesis |
| id | oxford-uuid:564cce16-1ee2-4cbf-8e49-daf0bb39ee36 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:10:40Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | oxford-uuid:564cce16-1ee2-4cbf-8e49-daf0bb39ee362022-06-29T12:22:20ZThree problems in additive combinatoricsThesishttp://purl.org/coar/resource_type/c_db06uuid:564cce16-1ee2-4cbf-8e49-daf0bb39ee36MathematicsEnglishHyrax Deposit2021Wang, RSanders, T<p>The problems considered in this thesis involve several fundamental objects in additive combinatorics. In this project, we mainly focus on the structure of subsets of integers. As integers are the most fundamental objects in number theory, one of our main interests in this project is to understand additive combinatorial problems related to sets which arise naturally in number theory, such as prime numbers and smooth numbers.</p> <p>In Chapter 1, we study the size of subsets of smooth numbers which are free of geometric progressions. By taking a Freiman isomorphism, we shall use additive results to study the multiplicative structure in smooth numbers. Our result is proved in Section 1.2 and Section 1.3. For completeness, we illustrate a typical counterexample in Section 1.4, namely the set of <em>k</em>-free smooth numbers. </p> <p>In the second problem, we focus on difference sets which do not contain shifted primes. Let ℙ denote the set of the prime numbers. Suppose <em>A</em> ⊆ [<em>N</em>] is nonempty and it satisfies (<em>A-A</em>) ∩ (ℙ - 1) = ∅. We are primarily interested in estimating the cardinality of the set <em>A</em> In Chapter 2, we prove a new upper bound on the cardinality of <em>A</em>. In Chapter 3, we investigate sets <em>A</em> such that <em>nA</em> - <em>nA</em> avoids ℙ - 1. We first prove an upper bound on |<em>A</em>| which depends on <em>n</em>, then we generalise an example of Ruzsa.</p> <p>In the third problem, we are interested in the quantitative aspect of partition regularity of the equation <em>x</em>+<em>y</em> = <em>z</em> when <em>x,y,z</em> are restricted to shifted primes ℙ - 1. In Chapter 4, we provide an exposition of the work of Li and Pan, which is the first result concerning partition regularity over shifted primes. We focus on the analytical ingredients of their transference principle and the quantitative bound that follows from it. In Chapter 5, we improve the bound produced by the method of Li and Pan using a different approach. The main ingredient of the proof is an iteration lemma from Chapter 2.</p> |
| spellingShingle | Mathematics Wang, R Three problems in additive combinatorics |
| title | Three problems in additive combinatorics |
| title_full | Three problems in additive combinatorics |
| title_fullStr | Three problems in additive combinatorics |
| title_full_unstemmed | Three problems in additive combinatorics |
| title_short | Three problems in additive combinatorics |
| title_sort | three problems in additive combinatorics |
| topic | Mathematics |
| work_keys_str_mv | AT wangr threeproblemsinadditivecombinatorics |