Borys Kuca

A celebrated theorem of Szemerédi states that each dense subset of integers contains an arithmetic progression of arbitrary length. Its polynomial extension by Bergelson and Leibman establishes the same conclusion for integral polynomial progressions. A lot of work has recently been done to quantify the latter result. My contribution consists of the following papers.

Bounds in the integers:

1. Corners with polynomial side length, with N. Kravitz and J. Leng, arXiv:2407.08637, arxiv, slides, Noah's video.

2. Quantitative concatenation for polynomial box norms, with N. Kravitz and J. Leng,  arXiv:2407.08636, arxiv, slides.

Bounds over the finite fields:

3. Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control, Adv. Math., 448:109700, 2024, journal, arxiv.

4. Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees, Israel J. Math. 259 (2024), 589-620, journal, arxiv.

5. Further bounds in the polynomial Szemerédi theorem in finite fields,  Acta Arith. 198 (2021), 77-108, journal, arxiv.

Bounds in the continuous setting:

6. On a continuous Sárközy type problem, with T. Orponen and T. Sahlsten, Inter. Math. Res. Not. IMRN, 2022;, rnac168, journal, arxiv, slides.

Finding bounds for avoiding a given polynomial progression is related to determining the complexity of the progression, which encapsulates the minimal toolbox that one needs to study the progression. There are several notions of complexity, coming from higher order Fourier analysis (true complexity), ergodic theory (Host-Kra and Weyl complexities) and algebra (algebraic complexity). All these notions are conjectured to agree. I am proving partial results towards this conjecture in the following two papers. 

1. On several notions of complexity of polynomial progressions, Ergodic Theory Dynam. Systems, 2023;43(4):1269-1323, journal, arxiv, slides (easy version), slides (advanced version), video.

2. True complexity of polynomial progressions in finite fields, Proc. Edinb. Math. Soc., 64.3 (2021), 448-500, journal, arxiv. 

Quantitative results in this direction also follow from my papers on the bounds in the polynomial Szemerédi theorem.

A central theme in ergodic theory is to understand the limiting behaviour of multiple ergodic averages that come from the ergodic proof of Szemerédi's theorem and its generalizations. I am exploring this problem in the following papers. 

1. Resolving the joint ergodicity problem for Hardy sequences, with S. Donoso, A. Koutsogiannis, W. Sun and K. Tsinas, arXiv:2506.20459, arxiv.

2. Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences, with S. Donoso, A. Koutsogiannis, W. Sun and K. Tsinas, arXiv:2410.15130, arxiv.

3. Joint ergodicity for commuting transformations and applications to polynomial sequences, with N. Frantzikinakis, Invent. Math., 239, 621–706 (2025), journal, arxiv. slides, video, Nikos' video .

4. Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates, with N. Frantzikinakis, Ergodic Theory Dynam. Systems, 2023;43(12):4074-4137, journal, arxiv.

5. Degree lowering for ergodic averages along arithmetic progressions, with N. Frantzikinakis,  J. Anal. Math., 2024, journal, arxiv, slides, video.

6. On several notions of complexity of polynomial progressions, Ergodic Theory Dynam. Systems, 2023;43(4):1269-1323, journal, arxiv, slides (easy version), slides (advanced version), video.

We say that a family of integer sequences is jointly ergodic for a given dynamical system if the corresponding multiple ergodic average converges in norm to the product of integrals. Joint ergodicity has been studied for a variety of sequences, including linear sequences, polynomials, generalised linear functions, Hardy field sequences, fractional powers of primes, etc. My contributions to this topic involve the following papers.

1. Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences, with S. Donoso, A. Koutsogiannis, W. Sun and K. Tsinas, arXiv:2410.15130, arxiv.

2. Joint ergodicity for commuting transformations and applications to polynomial sequences, with N. Frantzikinakis, Invent. Math., 239, 621–706 (2025), journal, arxiv. slides, video, Nikos' video .

3. Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates, with N. Frantzikinakis, Ergodic Theory Dynam. Systems, 2023;43(12):4074-4137, journal, arxiv.

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 mysterious phenomena that we have yet to explain. 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, J. Number Theory 194 (2019), 409-425, journal, slides.

2. Structures in additive sequences, Acta Arith. 186.3 (2018), 273-300, journal, arxiv.

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