# Borys Kuca

Cześć! I am a PhD student at University of Manchester, coming originally from Poland. Previously, I was studying at Yale University. I work under Sean Prendiville, Tuomas Sahlsten and Donald Robertson, and my research concerns additive combinatorics, with links to number theory and ergodic theory.

Here is my CV.

A complete list of my publications and preprints is available here.

Below, you can find a short description of my research.

## Combinatorial and dynamical properties of polynomial progressions

An integral polynomial progression is a configuration of the form x, x + P_1(y), ..., x + P_t(y) for polynomials P_1, ..., P_t with integer coefficients. Examples include

• arithmetic progressions x, x + y, ..., x + m y

• shifted geometric progressions x, x + y, ..., x + y^m

• x, x + y, x + y^2, x + y + y^2

• x, x + y, x + 2y, x + y^2.

There are many questions we can ask about polynomial progressions. For instance,

1. How big can a subset A of {1, ..., N} or F_p be if A lacks a given progression?

2. How many progressions of a given form does an arbitrary subset A of {1, ..., N} or F_p have?

3. What is the "complexity" of a given progression?

My papers below tackle these questions for certain progression, such as x, x+y, x+2y, x+y^3 or x, x+y^2, x+2y^2:

1. True complexity of polynomial progressions in finite fields, accepted by the Proceedings of the Edinburgh Mathematical Society.

2. Further bounds in the polynomial Szemerédi theorem in finite fields, Acta Arithmetica 198 (2021), 77-108.

## Ulam sequences

Let a < b be positive integers. We construct the Ulam sequence U(a,b) by starting with a and b, and then taking each subsequent term to be the smallest natural number greater than the previous term that be expressed as a sum of some two distinct previous terms in exactly one way. For instance,

U(1,2) = 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, ...

Despite their seemingly simple definitions, Ulam sequences exhibit a number of weird phenomena that we still try to understand. This post by Stefan Steinerberger explains them well, and has a lot of great pictures on the subject.

My papers on Ulam sequence and related topics are listed below. Some of them were written together with J. Hinman, A. Schlesinger and A. Sheydvasser.

1. The unreasonable rigidity of Ulam sets, with J. Hinman, A. Schlesinger, and A. Sheydvasser, Journal of Number Theory 194 (2019), 409-425.

2. Structures in additive sequences, Acta Arithmetica 186.3 (2018), 273-300.

3. Rigidity of Ulam sets and sequences, with J. Hinman, A. Schlesinger, and A. Sheydvasser, Involve 12-3 (2019), 521-539.