Related papers: Furstenberg set problem and exceptional set estima…

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to…

Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the M\"{o}bius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters.…

Field theoretical models with first order Lagrangean can be formulated in a covariant Hamiltonian formalism. In this article, the geometrical construction of the Gerstenhaber structure that encodes the equations of motion is explained for…

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean…

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite…

We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence results, they outperform their more general counterparts in…

In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear…

This work deals with the presence of localized structures in relativistic systems described by two real scalar fields in two-dimensional spacetime. We consider the usual two-field model with the inclusion of the cuscuton term, which couples…

The counting and (upper) mass dimensions are notions of dimension for subsets of $\mathbb{Z}^d$. We develop their basic properties and give a characterization of the counting dimension via coverings. In addition, we prove Marstrand-type…

We consider a Bertrand type estimate for primes splitting completely. As one of its applications, we show the finiteness of trivial solutions of Diophantine equation about the factorial function over number fields except for the case the…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

Canonical analysis leading to formal quantisation of the higher derivative theories are considered. The first order formalism is adopted where all the configuration space variables along with their higher time derivatives are considered to…

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns with the concept of multiple peaks. A theorem of Chatterjee states that when a centered Gaussian field admits the so-called…

The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers…

The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly…

We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ \Omega=\{\xi\in \mathbb{R}: \dim_B(E+\xi E)=\dim_B E\}. $$ We prove that either $\Omega=\{0\}$ or $\Omega$ is a…

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…

We apply the theory of Peres and Schlag to obtain estimates for generic Hausdorff dimension distortion under orthogonal projections on simply connected two dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain…

In this note we address the relation between symbolic and ordinary powers of the ideal of a reduced set or points in projective space: the so-called containment problem. In particular, we obtain sharp lower bounds on the Waldschmidt…