Zero‐one Laws in Simultaneous and Multiplicative Diophantine Approximation

Abstract

Abstract. Answering two questions of Beresnevich and Velani, we develop zeroone laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani. IntroductionDiophantine approximation is the quantitative study of rational number approximation to real numbers, originating from the celebrated theorem of Dirichlet that for any irrational number α, there exist infinitely many (q, p) ∈ N × Z satisfying |α − ) often referred to as an approximation function, a long-standing conjecture of Duffin and Schaeffer ([10]) says thatwhere M 1 (·) denotes the one-dimensional Lebesgue measure,In this paper we will study a specific question of such flavour, that is, the so-called zero-one law in Diophantine approximation. Given an approximation function ψ, Cassels ([9]) proved that the measure of the set of real numbers x in I [0, 1] for which |qx − p| < ψ(q) holds for infinitely many pairs of positive integers (q, p) equals either 0 or 1, while Gallagher ([11]) showed M 1 (lim sup q→∞ E q (ψ)) ∈ {0, 1}. The Cassels-Gallagher theorems called also zero-one laws have played a fundamental role in the study of metric Diophantine approximation (see e.g. [14,19]).2000 Mathematics Subject Classification. 11J13, 11J83.

The Duffin–schaeffer‐type Conjectures in Various Local Fields

“…[5,7,26]) as we have the cross fibering principle due to Beresnevich, Haynes and Velani ( [5]). Many conjectures in metric number theory were formulated to prove a particularly chosen {F n } n having the Duffin-Schaeffer property (see e.g.…”

“…Many conjectures in metric number theory were formulated to prove a particularly chosen {F n } n having the Duffin-Schaeffer property (see e.g. [2,5,26]). Hence Theorem 3.1 provides a new way to look at such kind of conjectures.…”

The Duffin-Schaeffer type conjectures in various local fields

“…Note in the simultaneous and multiplicative Diophantine approximation the zeroone property is in general not a big problem (see e.g. [5,7,26]) as we have the cross fibering principle due to Beresnevich, Haynes and Velani ( [5]). Many conjectures in metric number theory were formulated to prove a particularly chosen {F n } n having the Duffin-Schaeffer property (see e.g.…”

“…The proof of Lemma 7.4 is similar to those of [24,Thm. 4] and [26,Lemma 2.1] with suitable modifications, and we leave the details to the interested readers. Now we can give a proof of Lemma 7.2 and will only deal with the case d = 2 for the sake of simplicity.…”

A note on the weighted Khintchine-Groshev Theorem

“…The proof of Lemma 2 follows on combining Theorem 4 of [6] and Lemma 2.2 of [17] as described in [6]. It can also be proven using the "cross-fibering principle" described in [3], which allowed the authors to establish a ZeroOne Law in the multiplicative setup.…”