arXiv subjects
Publications and source records attributed to Noy Soffer Aranov.
We prove a quantitative version of Oppenheim's conjecture in the function field setting. In order to do so, we compute the higher moments of the Siegel transform. In particular, we find an optimal bound on the variance of the number of lattice points in a set. Moreover, we compute the exact variance of the number of lattice points in a ball, which is of independent interest.
Given a subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$ and fixed integers $n,m\in \mathbb{N}$, we study the distribution of the smallest denominator $Q\in \mathcal{S}$ for which there exists $\mathbf{P}\in \mathbb{F}_q[x]^m$ such that $\left\Vert\frac{\mathbf{P}}{Q}-\boldsymbolα\right\Vert<q^{-n}$, where $\boldsymbolα\in x^{-1}\mathbb{F}_q((x^{-1}))^m$ is chosen randomly. We also consider the discrete analogue obtained by fixing a polynomial $N\in \mathbb{F}_q[x]$ with $°(N)=n$ and sampling $\boldsymbolα$ uniformly from $\frac{1}{N}\mathbb{F}_q[x]^m$. We prove that for any infinite subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$, for every $n\in \mathbb{N}$ and every dimension $m$, the probability distributions of these two random variables coincide. This result is significantly stronger than the corresponding statement in the real setting, where Balazard and Martin showed that the averages of the discrete and continuous smallest denominator functions are asymptotically close.
We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. To address the scenario where the given approximating function is not monotonic, we extend our function field by adjoining an appropriate root. Additionally, in the course of the proof, we establish several new results in the geometry of numbers over function fields, which are of independent interest.
Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of Soffer-Aranov and Behajaina on counting problems related to orthogonality in $\mathcal{K}^n$. For example, we resolve an open question posed in Soffer-Aranov and Behajaina by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we also use ultrametric orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima.
Periodic lattices are natural generalizations of lattices, which arise naturally in diophantine approximations with rationals of bounded denominators. In this paper, we prove analogues of classical theorems in geometry of numbers for periodic lattices in function fields. Moreover, we use special matrices to compute the covering and packing radii of special periodic lattices.
Let $p$ be a prime. In 2017, Kemarsky, Paulin, and Shapira (KPS) conjectured that any Laurent series over $\mathbb{F}_p$ exhibits full escape of mass with respect to any irreducible polynomial $P(t)\in\mathbb{F}_p[t]$. In 2025, this was shown to be false in the case $p=2$ and $P(t)=t$ by Nesharim, Shapira and the first named author. This work shows that for any odd prime $p$ and any irreducible polynomial $P(t)\in\mathbb{F}_p[t]$, the so-called $p$-Cantor sequence provides a counterexample to the aforementioned conjecture over $\mathbb{F}_p$. Furthermore, the concepts of maximal escape of mass and generic escape of mass are introduced. These lead to two natural variations of the KPS conjecture, both of which are shown to hold for all previous counterexamples.
This is the first of a pair of papers, whose collective goal is to disprove a conjecture of Kemarsky, Paulin, and Shapira (KPS) on the escape of mass of Laurent series. This paper lays the foundations on which its sibling builds. In particular, the $p$-Cantor sequence is introduced. This generalises the classical Cantor sequence into a $p$-automatic sequence for any odd prime $p$. Two main results are then established, both of which play a key role in the disproof of the KPS conjecture. First, the two-dimensional sequence comprised of the Toeplitz determinants of the $p$-Cantor sequence over $\mathbb{F}_p$ is extensively studied. Indeed, the so-called profile of this sequence (which encodes the zero regions) is shown to be [p,p]-automatic. In the process of deriving this, the theory of so-called number walls is developed greatly. Many of these results are stated in full generality, as the authors expect them to be useful when tackling similar problems going forward. Secondly, a natural process is described that converts number wall of an automatic sequence into a unique fractal. When this sequence is the aforementioned $p$-Cantor sequence, this fractal is shown to have Hausdorff dimension $\log((p^2+1)/2)/\log(p).$
We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the the random variable which returns the degree of the smallest denominator $Q$, for which the ball of a fixed radius around a point contains a rational function of the form $\frac{P}{Q}$. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and $P$-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.
Every Laurent series in $\mathbb{F}_q\left(\left(t^{-1}\right)\right)$ has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teulié proved that the degrees of the partial quotients of the left-shifts of every quadratic Laurent series are unbounded. Shapira and Paulin and Kemarsky improved this by showing that certain sequences of probability measures on the space of lattices in the plane $\mathbb{F}_q\left(\left(t^{-1}\right)\right)^2$ exhibit positive escape of mass and conjectured that this escape is full -- that is, that these probability measures converge to zero. We disprove this conjecture by analysing in detail the case of the Laurent series over $\mathbb{F}_2$ whose sequence of coefficients is the Thue-Morse sequence. The proof relies on the discovery of explicit symmetries in its number wall.
We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier [Duke Math. J. 165 (2016), 2273--2329].
In this paper, we shall discuss topics in geometry of numbers in the positive characteristic setting, such as covering radii. We find a closed form for covering radii with respect to convex bodies, which will lead to a proof of the positive characteristic analogue of Woods' conjecture in this setting. Then, we will prove a positive characteristic analogue of Minkowski's conjecture about the multiplicative covering radius. To do this, we shall prove a positive characteristic analogue of Solan's result that every diagonal orbit intersects the set of well rounded lattices. This implies that the Gruber-Mordell spectrum in positive characteristic is trivial in every dimension.
Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper, motivated by a question of Erd{ő}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ satisfying the following property: for any $E \subseteq S$ of size $k$, there exists $F \subseteq E$ of size $\ell$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied.
We show that there are infinitely many counterexamples to Minkowski's conjecture in positive characteristic regarding uniqueness of the upper bound of the multiplicative covering radius, $μ$, by constructing a sequence of compact $A$ orbits where $μ$ obtains its conjectured upper bound. In addition, we show that these orbits, as well as a slightly larger sequence of orbits, must exhibit complete escape of mass.