[
    {
        "id": "thesis:17226",
        "collection": "thesis",
        "collection_id": "17226",
        "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05132025-055003057",
        "type": "thesis",
        "title": "Sums of Various Dilates",
        "author": [
            {
                "family_name": "Lim",
                "given_name": "Jeck",
                "orcid": "0009-0002-0369-8523",
                "clpid": "Lim-Jeck"
            }
        ],
        "thesis_advisor": [
            {
                "family_name": "Conlon",
                "given_name": "David",
                "orcid": "0000-0001-5899-1829",
                "clpid": "Conlon-David"
            }
        ],
        "thesis_committee": [
            {
                "family_name": "Hutchcroft",
                "given_name": "Thomas",
                "orcid": "0000-0003-0061-593X",
                "clpid": "Hutchcroft-Tom"
            },
            {
                "family_name": "Conlon",
                "given_name": "David",
                "orcid": "0000-0001-5899-1829",
                "clpid": "Conlon-David"
            },
            {
                "family_name": "Dimitrov",
                "given_name": "Vesselin",
                "clpid": "Dimitrov-Vesselin"
            },
            {
                "family_name": "Pham",
                "given_name": "Huy Tuan",
                "clpid": "Pham-Huy-Tuan"
            }
        ],
        "local_group": [
            {
                "literal": "div_pma"
            }
        ],
        "abstract": "<p>Given a finite subset A of an ambient abelian group and a dilate \u03bb, how large must the sum of dilate A+\u03bb\u2219A be in terms of A? In this thesis, we study this problem in various settings and generalizations, proving tight bounds in many cases. Our five main results are as follows.</p>\r\n\r\n<p>1. In the setting of a d-dimensional subset A of \u211d\u1d48, we prove an exact lower bound on the size of the difference set A-A.</p>\r\n\r\n<p>2. In the case when \u03bb \u2208 \\in C is a transcendental number, we show that there is an absolute constant c&gt;0 such that |A+\u03bb\u2219A|\u2265 \r\n exp(c\u221alog|A|)|A| for any finite subset A of C. This is best possible up to the constant c.</p>\r\n\r\n<p>3. In the algebraic case, given algebraic numbers \u03bb<sub>1</sub>,...,\u03bb<sub>k</sub>, we prove tight lower bounds for the sum of dilates A+\u03bb\u2219A+ ... \u03bb<sub>k</sub>\u2219A. As an important ingredient, we also prove a Freiman-type structure theorem for sets with small sums of dilates.</p>\r\n\r\n<p>4. In the setting of sums of linear transformations, we prove tight bounds for the sum of two linear transformations and tight bounds for the sum of multiple pre-commuting linear transformations.</p>\r\n\r\n<p>5. In the setting of groups of prime order, we prove near-optimal lower and upper bounds for the sum of dilate A+\u03bb\u2219A for A of a given density and large \u03bb.</p>",
        "doi": "10.7907/6q8b-ck71",
        "publication_date": "2025",
        "thesis_type": "phd",
        "thesis_year": "2025"
    }
]