The most versatile version of the classical divergence Borel-Cantelli lemma shows that for
any divergent sequence of events $ E_n $ in a probability space satisfying a quasi …

Abstract Let (Ω, A, μ) be a probability space. The classical Borel–Cantelli Lemma states that
for any sequence of μ-measurable sets E i (i= 1, 2, 3,…), if the sum of their measures …

How often is a quintic polynomial solvable by radicals? We establish that the number of such
polynomials, monic and irreducible with integer coefficients in $[-H, H] $, is $ O (H^{3.91}) …

Let ε> 0. We construct an explicit, full-measure set of α∈[0, 1] such that if γ∈ R then, for
almost all β∈[0, 1], if δ∈ R then there are infinitely many integers n⩾ 1 for which n‖ n α …

We determine the generic multiplicative approximation rate on a hypersurface. There are
four regimes, according to convergence or divergence and curved or flat, and we address all …

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in
dimension $ m\geq 3$. That is, given $\mathbf {y}\in\mathbb {R}^ m $ and $\psi:\mathbb …

In this paper, we prove an effective asymptotic equidistribution result for one-parameter
unipotent orbits in $\mathrm {SL}(3,\mathbb {R})/\mathrm {SL}(3,\mathbb {Z}) $. This enables …

The Diophantine sums $\sum_ {n= 1}^ N\| n\alpha\|^{-1} $ and $\sum_ {n= 1}^ N n^{-1}\|
n\alpha\|^{-1} $ appear in many different areas including the ergodic theory of circle …

arXiv:2412.12070v1 [math.NT] 16 Dec 2024 Page 1 arXiv:2412.12070v1 [math.NT] 16 Dec
2024 SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON …

In this paper, we answer a question of Cai-Hambrook in (arXiv $\colon $2403.19410).
Furthermore, we compute the Fourier dimension of the multiplicative $\psi $-well …