Borys Kuca



Without a proof, you are just another person with an idea.

Cześć! I am a postdoctoral research at Jagiellonian University in Kraków, Poland, and the principal investigator of the Polonez Bis 3 grant "Ergodic theory meets combinatorial number theory".


I obtained my undergraduate degree from Yale University and completed my PhD at University of Manchester under the supervision of Sean Prendiville, Tuomas Sahlsten and Donald Robertson. After that, I was a postdoc in Jyväskylä (Finland) and Crete (Greece). My research concentrates on ergodic theory and additive combinatorics, with links to number theory and fractal geometry. 


Here is my CV, and a complete list of my publications and preprints is available here.


As a former champion in Tarnów Debate League and the Polish Championship of Oxford-style debates, I am co-authoring a blog (in Polish) on debates, rhetoric and logical thinking. 


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

Bounds for sets lacking polynomial progressions

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. There has been a lot of research in the last decades to find good bounds for the size of subsets of integers or finite fields lacking a given polynomial configuration, and my contribution to this subject consists of the following four papers:

In (1), I prove upper bounds for the size subsets of finite fields lacking certain single-dimensional polynomial configurations. In (2) and (3), I examine the same question for multidimensional progressions coming from distinct-degree and linearly independent polynomials respectively. In (4), together with Tuomas Orponen and Tuomas Sahlsten, we study an analogous question in the fractal setting. 

Complexity of polynomial progressions

Finding bounds for subsets of integers or finite fields lacking a given polynomial progression is related to determining the complexity of the progression. There are several notions of complexity, coming from higher order Fourier analysis, ergodic theory and algebra. The true complexity captures the level of uniformity that a set has to satisfy so that it contains roughly as many progressions of a certain type as a random set of the same cardinality would. The Host-Kra and Weyl complexity describe the smallest factor of a dynamical system on which certain multiple ergodic averages can be projected without a change to their limits. It is conjectured that all these notions agree, and that the true, Host-Kra or Weyl complexities of a progression depend solely on the algebraic relations satisfied by the terms of the progression. I am proving partial results towards this conjecture in the following two papers. 

Some results in this direction also follow from the paper below.

Limiting behaviour of multiple ergodic averages

The polynomial Szemerédi theorem by Bergelson and Leibman guarantees that each dense subset of natural numbers contains certain polynomial patterns, extending the theorem of Szemerédi on arithmetic progressions. Its proof comes from ergodic theory, and it relies on establishing the positivity of a liminf associated with a certain family of multiple ergodic averages. Understanding the limiting behaviour of such averages has become an active area of research ever since. We know that multiple ergodic averages with polynomial iterates converge in norm thanks to works of Host and Kra, Leibman, Tao, Walsh, Zorin-Kranich and others. But can we say anything more about the limit? I am exploring this problem in the following three papers. 

In the first paper, Nikos Frantzikinakis and I establish necessary and sufficient criteria for joint ergodicity of a general family of integer sequences with respect to a system of commuting transformations. We then combine it with a novel seminorm smoothing technique to obtain a number of results for averages with polynomial iterates, including seminorm control, optimal characteristic factors and limiting formula. In the second paper, we refine the techniques to cover the case of not necessarily distinct polynomials. In the third paper, we give a new set of necessary and sufficient criteria for when a Host-Kra factor of some fixed degree is characteristic for a certain ergodic average, with applications to arithmetic progressions with restricted differences. In the fourth paper, I am examining averages of a single transformation with polynomial iterates, finding the smallest Host-Kra factor characteristic for a large family of such averages. 

Joint ergodicity of sequences

We say that a family of integer sequences is jointly ergodic for a given dynamical system if the corresponding multiple ergodic average converges to the product of integrals in norm. 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 two papers written jointly with Nikos Frantzikinakis.

In the first paper, we establish necessary and sufficient criteria for joint ergodicity of a general family of integer sequences with respect to a system of commuting transformations. We then combine this with a novel seminorm smoothing technique to prove the joint ergodicity of linearly independent polynomials. We also provide a spectral criterion for joint ergodicity of any sequence of essentially distinct, pairwise independent  family of polynomials. In the second paper, we refine the techniques to cover the case of not necessarily distinct polynomials. 

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 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.