Arithmetic Combinatorics
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THOMAS BLOOM, University of Bristol
Mixed additive structure and applications  [PDF]

In their breakthrough work on the cap set problem, Bateman and Katz introduced a new type of structural result in arithmetic combinatorics, capable of decomposing a wide variety of sets into a structured and pseudorandom component. I will discuss some extensions of this result and applications, including quantitative bounds for Roth's theorem.

NEIL LYALL, University of Georgia
Geometric Ramsey Theory  [PDF]

I will discuss some recent results in Geometric Ramsey Theory.

ALEX RICE, University of Rochester
Extending the Best Known Bounds for the Furstenberg-S\'ark\"ozy Theorem  [PDF]

It is a well known result, established independently by S\'ark\"ozy and Furstenberg, that a set of integers with positive upper density must contain two distinct elements that differ by a perfect square. The best-known quantitative upper bounds for this result were established with an intricate Fourier analytic argument by Pintz, Steiger, and Szemer\'edi. In this talk, we discuss the extension of these bounds from perfect squares to the largest possible class of polynomials, as well as other related results.

GEORGE SHAKAN, University of Illinois at Urbana-Champaign
On the sum of dilations of a set  [PDF]

Let $A \subset \mathbb{Z}$ be finite and nonempty and $q \geq 2$ be an integer. Utilizing only elementary techniques, we showed that $$|A+q \cdot A| \geq (q+1)|A| - q^{(q-2)(q+1) +1}.$$ A key step of our proof is to introduce a paramater $1 \leq m \leq q+1$ and show $|A+q \cdot A| \geq m|A| + O_m(1)$, by inducting on $m$ (using that the result holds for $m$ to prove it for $m+1/q$). We plan to outline some of these ideas and talk about related results and open problems. For example, the natural higher dimensional analog of the above is currently open.

ANA ZUMALACÁRREGUI, University of New South Wales
Threshold functions and Poisson convergence for systems of equations in random sets  [PDF]

We will study the existence of solutions to linear systems of equations in random sets and define a unified framework which includes AP's, sum-free~sets, $B_{h}[g]$-sets or Hilbert~cubes.

In particular, we will show the existence of a threshold function for the property \textit{$A$ contains a non-trivial solution $\ M\cdot x = 0$'' } where $A$ is a random set of $[n]$. Furthermore, we show that the number of solutions in the threshold scale converges to a Poisson distribution whose parameter depends on the volumes of certain polytopes arising from the system under study.

Joint work with J.~Ru\'{e} and C.~Spiegel.