CaltechTHESIS committee: Video
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 06 Aug 2024 18:54:00 -0700Extremal Results in and out of Additive Combinatorics
https://resolver.caltech.edu/CaltechTHESIS:05232020-121812350
Year: 2020
DOI: 10.7907/mb11-pg80
<p>In this thesis, we study several related topics in extremal combinatorics, all tied together by various themes from additive combinatorics and combinatorial geometry.</p>
<p>First, we discuss some extremal problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local property to derive global properties of the entire configuration. We study one such Ramsey problem of Erdős and Shelah, where the configurations are complete graphs with colored edges and every
small induced subgraph contains many distinct colors. Our bounds for this Ramsey
problem show that the known probabilistic construction is tight in various cases. We
study one discrete geometry variant, also by Erdős, where we have a set of points in the
plane such that every small subset spans many distinct distances. Finally, we consider
an arithmetic variant, suggested by Dvir, where we are given sets of real numbers such that
every small subset has a large difference set. In Chapter 2, we derive new bounds for all of the above problems. Along the way, we also essentially answer a question of Erdős and Gyárfás.</p>
<p>Second, we study the behavior of expanding polynomials on sets with additive or multiplicative structure. Given an arbitrary set of real numbers <i>A</i> and a two-variate polynomial <i>f</i> with real coefficients, a remarkable theorem of Elekes and Rónyai from 2000 states that the size |<i>f</i>(<i>A</i>,<i>A</i>)| of the image of <i>f</i> on the cartesian product <i>A</i> × <i>A</i> grows asymptotically faster than |<i>A</i>|, unless <i>f</i> exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the <i>sum-product problem</i> in additive combinatorics. In Chapter 3, we discuss new bounds for |<i>f</i>(<i>A</i>,<i>A</i>)| when the set <i>A</i> has few sums or few products.</p>
<p>Another central problem in additive combinatorics is the problem of finding good quantitative bounds for maximal progression-free sets in the integers (or various other groups). In 2017, a major breakthrough of Croot, Lev and Pach took the community by surprise with impressive new bounds for the problem in ℤ<sub>4</sub><sup><i>n</i></sup> and in higher order 2-abelian groups. Their new polynomial method was quickly adapted by Ellenberg and Gijswijt to show a similar strong result for the size of the largest three-term progression free subset of 𝔽<sub><i>q</i></sub><sup><i>n</i></sup> where <i>q</i> is an odd prime power, the so-called <i>cap set problem</i>. This new set of ideas has subsequently led to very exciting developments in a vast range of topics. The rest of the thesis will be dedicated to discussing my joint results around these new developents. In Chapter 4, we develop a new multi-layered polynomial method approach to derive improved bounds for the largest three-term progression free set in ℤ<sub>8</sub><sup><i>n</i></sup> (which also improve on the Croot-Lev-Pach bounds for a large family of higher order 2-abelian groups). In Chapter 5, we generalize the Ellenberg-Gijswijt bound for the cap set problem to random progression-free subsets of 𝔽<sub><i>q</i></sub><sup><i>n</i></sup>, improving on a theorem of Tao and Vu. A result of this type enables one to find four term progressions-free sets which contain three-term progressions in all of their large subsets (with good quantitative bounds), but which do not contain too many three-term progressions overall. Motivated by this application, in Chapter 6 we continue this investigation and study further the question of determining the maximum total number of 3APs in a given 4AP-free set. We show in general, for all fixed integers <i>k</i> > <i>s</i> ≥ 3, that if <i>f<sub>s,k</sub></i>(<i>n</i>) denotes the maximum possible number of <i>s</i>-term arithmetic progressions in a set of <i>n</i> integers which contains no <i>k</i>-term arithmetic progression, then <i>f<sub>s,k</sub></i>(<i>n</i>) = <i>n</i><sup>2-<i>o</i>(1)</sup>. This answers an old question of Erdős. In Chapter 7, we study some limitations of the Croot-Lev-Pach approach and discuss some problems at the intersection of extremal set theory and combinatorial geometry where one can use additional linear algebraic ideas to go slightly beyond the Croot-Lev-Pach method.</p>https://resolver.caltech.edu/CaltechTHESIS:05232020-121812350Descriptive Set Theory and Dynamics of Countable Groups
https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796
Year: 2022
DOI: 10.7907/egch-kp69
<p>This thesis comprises four papers.</p>
<p> 1. We show that for any Polish group G and any countable normal subgroup Γ ⊳ G, the coset equivalence relation G/Γ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.</p>
<p> 2. Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X. </p>
<p> 3. Let Γ be a countable group. A classical theorem of Thorisson states that if X is a standard Borel Γ-space and µ and ν are Borel probability measures on X which agree on every Γ-invariant subset, then µ and ν are equidecomposable,
i.e., there are Borel measures (µ<sub>γ</sub>)<sub>γϵΓ</sub> on X such that µ = Σ<sub>γ</sub>µ<sub>γ</sub> and ν = Σ<sub>γ</sub>γµ<sub>γ</sub>. We establish a generalization of this result to cardinal algebras.</p>
<p> 4. Let R be a ring equipped with a proper norm. We show that under suitable conditions on R, there is a natural basis under continuous linear injection for the set of Polish R-modules which are not countably generated. When R is a division ring, this basis can be taken to be a singleton.</p>https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796On the Hecke Module of GLₙ(k[[z]])\GLₙ(k((z)))/GLₙ(k((z²)))
https://resolver.caltech.edu/CaltechTHESIS:12082023-083025167
Year: 2024
DOI: 10.7907/d0bn-5e47
<p>[See Abstract in text of thesis for correct representation of mathematics]</p>
<p>Every double coset in GLₘ(k[[z]])\GLₘ(k((z)))/GLₘ(k((z²))) is uniquely represented by a block diagonal matrix with diagonal blocks in { 1,z, (11 z \\0 zⁱ \\) (i>1) } if char(k) ≠ 2 and k is a finite field. These cosets form a (spherical) Hecke module H(G,H,K) over the (spherical) Hecke algebra H(G,K) of double cosets in K\G/H, where K=GLₘ(k[[z]]) and H=GLₘ(k((z²))) and G=GLₘ(k((z))). Similarly to Hall polynomial hλ,ν^µ from the Hecke algebra H(G,K), coefficients hλ,ν^µ arise from the Hecke module. We will provide a closed formula for hλ,ν^µ, under some restrictions over λ, ν, µ.</p>https://resolver.caltech.edu/CaltechTHESIS:12082023-083025167A Kakeya Estimate for Sticky Sets Using a Planebrush
https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252
Year: 2024
DOI: 10.7907/japt-b214
<p>A Besicovitch set is defined as a compact subset of ℝⁿ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d ⩾ 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called "stickiness." The improved estimate comes from using an incidence geometry argument called a "k-planebrush," which is a higher dimensional analogue of Wolff's "hairbrush" argument from 1995.</p>
<p>In addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in ℝⁿ must have Minkowski dimension greater than (2 - √2 + ε)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result.</p>https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252A Kakeya Estimate for Sticky Sets Using a Planebrush
https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252
Year: 2024
DOI: 10.7907/japt-b214
<p>A Besicovitch set is defined as a compact subset of ℝⁿ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d ⩾ 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called "stickiness." The improved estimate comes from using an incidence geometry argument called a "k-planebrush," which is a higher dimensional analogue of Wolff's "hairbrush" argument from 1995.</p>
<p>In addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in ℝⁿ must have Minkowski dimension greater than (2 - √2 + ε)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result.</p>https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252