arXiv subjects

Kamil Bulinski

Publications and source records attributed to Kamil Bulinski.

13 recordsLinked to original sources

arXiv2026

Pure or Unstable: A Generic Dichotomy for Strong Stackelberg Commitments

We study the robustness of the Strong Stackelberg Equilibrium (SSE) in finite leader--follower games when the follower's best-response correspondence is set-valued. While optimistic tie-breaking (in the leader's favor) is commonly adopted, it can hinge on knife-edge indifferences. We formalize a stability notion: an SSE is unstable if, at the leader's committed strategy, the follower has an alternative best response that strictly reduces the leader's payoff. Our main results establish a sharp generic dichotomy. Fixing the follower's utility and sampling the leader's utility from any continuous distribution, with probability one the optimal Stackelberg commitment is unique and is either (i) pure, or (ii) mixed and unstable. When both players' utilities are sampled generically, this strengthens to: with probability one, the unique optimal commitment is either pure and stable or mixed and unstable. These theorems complement the classic generic-value result of von Stengel and Zamir by showing that even when optimistic and pessimistic leader values coincide generically, the strategy-level SSE prediction is generically fragile whenever optimality requires genuine randomization. We further apply this perspective to Stackelberg satisfaction games, disproving a conjecture from prior work via counterexamples and identifying conditions under which it nonetheless holds.

cs.GT↗

arXiv2023

Counting elements of the congruence subgroup

We obtain asymptotic formulas for the number of matrices in the congruence subgroup \[ \Gamma_0(Q) = \left\{ A\in\mathrm{SL}_2(\mathbb Z):~c \equiv 0 \pmod Q\right\}, \] which are of naive height at most $X$. Our result is uniform in a very broad range of values $Q$ and $X$.

math.NT↗

arXiv2023

Counting embeddings of free groups into $\mathrm{SL}_2(\mathbb{Z})$ and its subgroups

We show that if one selects uniformly independently and identically distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the matrices $A_1, \ldots, A_s$ are free generators for a free subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Furthermore, to show the flexibility of our method we do similar counting for matrices from the congruence subgroup $\Gamma_0(Q)$ uniformly with respect to the positive integer $Q\le X$. This improves and generalises a result of E. Fuchs and I. Rivin (2017) which claims that the probability is $1 + o(1)$. We also disprove one of the statements in their work that has been used to deduce their claim.

math.NT↗

arXiv2022

Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $ε> 0$ there exists an integer $n$ such that $nY$ is $ε$-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on $\mathbb{T}^d$ also satisfy such a \textit{Glasner property} where each infinite set (in fact, arbitrarily large finite set) will have an $ε$-dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski-connected Zariski-closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.

math.DS↗

arXiv2021

Arithmetic subtrees in large subsets of products of trees

Furstenberg-Weiss have extended Szemerédi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend the multidimensional Szemerédi theorem by demonstrating the existence of certain arithmetic structures in large subsets of a cartesian product of trees.

math.CO↗

arXiv2021

Reconstructing a minimal topological dynamical system from a set of return times

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric closure, then that Kronecker system must be a factor. We also study similar problems involving Nilsystems and polynomial return times. We state a number of questions on whether these results extend to other homogeneous spaces and transitive group actions, some of which are already interesting for finite groups.

math.DS↗

arXiv2021

Glasner property for unipotently generated group actions on tori

A theorem of Glasner from 1979 shows that if $A \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $ε> 0$ there exists an integer $n$ such that $nA$ is $ε$-dense and Berend-Peres later showed that in fact one can take $n$ to be of the form $f(m)$ for any non-constant $f(x) \in \mathbb{Z}[x]$. Alon and Peres provided a general framework for this problem that has been used by Kelly-Lê and Dong to show that the same property holds for various linear actions on $\mathbb{T}^d$. We complement the result of Kelly-Lê on the $ε$-dense images of integer polynomial matrices in some subtorus of $\mathbb{T}^d$ by classifying those integer polynomial matrices that have the Glasner property in the full torus $\mathbb{T}^d$. We also extend a recent result of Dong by showing that if $Γ\leq \operatorname{SL}_d(\mathbb{Z})$ is generated by finitely many unipotents and acts irreducibly on $\mathbb{R}^d$ then the action $Γ\curvearrowright \mathbb{T}^d$ has a uniform Glasner property.

math.DS↗

arXiv2021

Quantitative twisted patterns in positive density subsets

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on random walks in certain subgroups of $\operatorname{SL}_r(\mathbb{Z})$ (the symmetry groups of these quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing that rather than studying random walks, one can obtain more quantitative results by considering polynomial orbits of these group actions that are not contained in cosets of submodules of $\mathbb{Z}^r$ of small index. Our main new technical tool is a uniform Furstenberg-S\'{a}rk\"{o}zy theorem that holds for a large class of polynomials not necessarily vanishing at zero, which may be of independent interest and is derived from a density increment argument and Hua's bound on polynomial exponential sums.

math.DS↗

arXiv2017

Twisted Recurrence via Polynomial Walks

In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying some irreducibility and polynomial assumptions. As one of the consequences, we prove a non-linear analog of Bogolubov's theorem -- for any set $B \subset \mathbb{Z}^2$ of positive density, and $p(n) \in \mathbb{Z}[n]$, with $p(0) = 0$ and $\operatorname{deg}(p) \geq 2$, there exists $k \geq 1$ such that $k \mathbb{Z} \subset \{ x - p(y) \, | \, (x,y) \in B-B \}$. Unlike the previous works on twisted recurrence that used recent results of Benoist-Quint and Bourgain-Furman-Lindenstrauss-Mozes on equidistribution of random walks on automorphism groups of tori, our method relies on the classical Weyl equidistribution for polynomial orbits on tori.

math.DS↗

arXiv2017

Spherical Recurrence and locally isometric embeddings of trees into positive density subsets of $\mathbb{Z}^d$

Magyar has shown that if $B \subset \mathbb{Z}^d$ has positive upper density $(d \geq 5)$, then the set of squared distances $\{ \|b_1-b_2 \|^2 \text{ }: \text{ } b_1,b_2 \in B \}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements with gaps in some given arithmetic progression (which depends on the length of the sought chain). This is a discrete analogue of a result obtained recently by Bennet, Iosevich and Taylor on chains with prescribed gaps in sets of large Haussdorf dimension. Our techniques are Ergodic theoretic and may be of independent interest to Ergodic theorists. In particular, we obtain Ergodic theoretic analogues of recent \textit{optimal spherical distribution} results of Lyall and Magyar which, via Furstenberg's correspondence principle, recover their combinatorial results.

math.DS↗

arXiv2017

Multidimensional lower density versions of Plünnecke's inequality

We investigate the lower asymptotic density of sumsets in $\mathbb{N}^2$ by proving certain Plünnecke type inequalities for various notions of lower density in $\mathbb{N}^2$. More specifically, we introduce a notion of lower tableaux density in $\mathbb{N}^2$ which involves averaging over convex tableaux-shaped regions in $\mathbb{N}^2$ which contain the origin. This generalizes the well known Plünnecke type inequality for the lower asymptotic density of sumsets in $\mathbb{N}$. We also provide a conjectural Plünnecke inequality for the more basic notion of lower rectangular asymtpotic density in $\mathbb{N}^2$ and prove certain partial results.

math.CO↗

arXiv2015

Twisted patterns in large subsets of $\mathbb{Z}^N$

Let $E \subset \mathbb{Z}^N$ be a set of positive upper Banach density and let $Γ< \operatorname{GL}_N(\mathbb{Z})$ be a finitely generated, strongly irreducible subgroup whose Zariski closure in $\operatorname{GL}_N(\mathbb{R})$ is a Zariski connected semisimple group with no compact factors. Let $Y$ be any set and suppose that $Ψ: \mathbb{Z}^N \rightarrow Y$ is a $Γ$-invariant function. We prove that for every positive integer $m$, there exists a positive integer $k$ with the property that for every finite set $F \subset \mathbb{Z}^N$ with $|F| = m$, we have \[ Ψ(kF) \subset Ψ(E-b) \quad \textrm{for some $b \in E$}. \] Furthermore, if $E$ is an aperiodic Bohr$_o$-set, we can choose $k = 1$ and $b = 0$. As one of many applications of this result, we show that if $E_o \subset \mathbb{Z}$ has positive upper Banach density, then, for any integer $m$, there exists an integer $k$ with the property for \emph{every} finite set $F \subset \mathbb{Z}$, we can find $x,y,z \in E_o$ such that \[ k^2 F \subset \big\{ (u-x)^2 + (v-y)^2 - (w-z)^2 \, : \, u,v,w \in E_o \big\}. \] In particular, if $E_o \subset \mathbb{Z}$ is an aperiodic Bohr$_o$-set, then every integer can be written on the form $u^2 + v^2 - w^2$ for some $u,v,w \in E_o$. Our techniques use recent results by Benoist-Quint and Bourgain-Furman-Lindenstrauss-Mozes on equidistribution of random walks on automorphism groups of tori.

math.DS↗

arXiv2014

Plünnecke inequalities for measure graphs with applications

We generalize Petridis's new proof of Plünnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Plünnecke inequalities for measure preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that improve on those given by Jin.

math.DS↗