相关论文: On the Mori-Szekely conjectures for the Borel-Cant…

In this short note, we discuss the Barndorff-Nielsen lemma, which is a generalization of well-known Borel-Cantelli lemma. Although the result stated in the Barndorff-Nielsen lemma is correct, it does not follow from the argument proposed in…

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

In the present note, we generalize the first part of the Borel-Cantelli lemma. By this generalization, we obtain some strong limit results.

In this short note, we briefly discuss the Borel-Cantelli lemma and propose a new generalization of the first part of it.

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

In the present note a generalization of Borel-Cantelli Lemma is proposed.

We give a version of the Borel-Cantelli lemma. As an application, we prove an almost sure local central limit theorem. As another application, we prove a dynamical Borel-Cantelli lemma for systems with sufficiently fast decay of…

This paper has been withdrawn by the author due to an error.

In this note, we discuss recently discovered counterexamples to Mordell's Pellian Equation Conjecture and the Ankeny-Artin-Chowla-Conjecture. We provide a verification of the counterexample to Mordell's Pellian Equation Conjecture that can…

The converse of the Borel-Cantelli Lemma states that if $\{A_i\}_{i=1}^\infty$ is a sequence of independent events such that $\sum P(A_i)=\infty$, then almost surely infinitely many of these events will occur. Erd\H os and R\'enyi proved…

We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.

Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel-Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding…

In this article, we give two different proofs of why the Collatz Conjecture is false.

We here aim to complete our model-theoretic account of the function field Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski geometries, where we now consider the general case of semiabelian varieties. The main…

In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem. We will see that the first three results have…

We study the difference between weak Morrey quasiconvexity and strong Morrey quasiconvexity in L^{\infty}. We point out some relations as well as give one example to show that weak Morrey quasiconvexity cannot imply strong Morrey…

Two conjectures recently proposed by one of the authors are disproved

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a…

Let $\{A_n\}_{n=1}^\infty$ be a sequence of events and let $\displaystyle S:=\sum_{n=1}^\infty 1_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P} (S<\infty)=1$ and $\mathbb{P} (S=\infty)=1$…

This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of…