Given a commutative unital ring R, we show that the finiteness length of a group G is bounded above by the finiteness length of the Borel subgroup of rank one $\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$ whenever G admits certain R-representations with metabelian image. Combined with results due to Bestvina–Eskin–Wortman and Gandini, this gives a new proof of (a generalization of) Bux’s equality on the finiteness length of S-arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups $\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$ in terms of n and $\textbf {B}_2^{\circ }(R)$ . This generalizes earlier results due to Remeslennikov, Holz, Lyul’ko, Cornulier–Tessera, and points out to a conjecture about the finiteness length of such groups.

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.

Involution Schubert polynomials represent cohomology classes of K-orbit closures in the complete flag variety, where K is the orthogonal or symplectic group. We show they also represent $\mathsf {T}$ -equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$ , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey–Jockusch–Stanley formula for Schubert polynomials. In Knutson and Miller’s approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

For fixed $\alpha \in [0,1]$ , consider the set $S_{\alpha ,N}$ of dilated squares $\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $ modulo $1$ . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such $\alpha $ the gap-distribution of $S_{\alpha ,N}$ is consistent with the Poisson model (in the limit as N tends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha ,N}$ in a random interval of length $L/N$ , provided that $L> N^{1/4+\varepsilon }$ . The threshold of $\tfrac {1}{4}$ is substantially smaller than the threshold of $\tfrac {1}{2}$ (which is the threshold that would be given by a naïve discrepancy estimate).

Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $ for a nonlacunary sequence $(a_n)_{n=1}^{\infty } $ of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.

In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

What is the maximum number of copies of a fixed forest T in an n-vertex graph in a graph class $\mathcal {G}$ as $n\to \infty $ ? We answer this question for a variety of sparse graph classes $\mathcal {G}$ . In particular, we show that the answer is $\Theta (n^{\alpha _{d}(T)})$ where $\alpha _{d}(T)$ is the size of the largest stable set in the subforest of T induced by the vertices of degree at most d, for some integer d that depends on $\mathcal {G}$ . For example, when $\mathcal {G}$ is the class of k-degenerate graphs then $d=k$ ; when $\mathcal {G}$ is the class of graphs containing no $K_{s,t}$ -minor ( $t\geqslant s$ ) then $d=s-1$ ; and when $\mathcal {G}$ is the class of k-planar graphs then $d=2$ . All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.

Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

Let G be a connected reductive group over a p-adic number field F. We propose and study the notions of G- $\varphi $ -modules and G- $(\varphi ,\nabla )$ -modules over the Robba ring, which are exact faithful F-linear tensor functors from the category of G-representations on finite-dimensional F-vector spaces to the categories of $\varphi $ -modules and $(\varphi ,\nabla )$ -modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show that G- $(\varphi ,\nabla )$ -modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of the p-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.

For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$ . Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$ , we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$ -symmetry.