Related papers: The $p$-adic Duffin--Schaeffer conjecture

In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a…

In this paper we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and…

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris…

For a non-zero algebraic number $\alpha$ of degree $d$, let $h(\alpha)$ denote its logarithmic Weil height. It is known that when $h(\alpha)$ is small, and $d$ is large, the conjugates of $\alpha$ are clustered near the unit circle and have…

With an artificial (p', n')-system it has been proved that even numbers > (p(x))^2 are the sum of two p > p(x).

In this paper we introduce the concept of an infinite loop mod $n$ and discuss the properties that these objects have. In particular, we show that a real number $\alpha$ is a counterexample to the $p$-adic Littlewood Conjecture if and only…

Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.

The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${\rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a…

This is an appendix to our paper "An update of the Hirsch Conjecture" (arXiv:0907.1186), containing proofs of some of the results and comments that were omitted in it.

We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by Dani, Laurent, and…

We prove Union-Closed sets conjecture.

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to…

One proves the Crew-Tsuzuki "p-adic local monodromy conjecture" (for local fields of characteristic p>0).

In this paper we prove three results conjectured by Z.-W. Sun. Let $p$ be an odd prime and let $h\in \mathbb{Z}$ with $2h-1\equiv0\pmod{p^{}}$. For $a\in\mathbb{Z}^{+}$ and $p^a>3$, we show that \begin{align}\notag…

We provide empirical evidence for the Erd\H{o}s-Straus conjecture by improving computational bounds to $10^{18}$ and by evaluating the solution-counting function $f(p)$ for this conjecture.

The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number.…

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class…

The article provides a counterexample to a conjecture by Blocki-Zwonek.

In this paper we investigate the recent advances by Zhang, Maynard and Pintz towards Polignac's conjecture and give some new results concerning the relationship between Polignac numbers and arithmetic progressions.