The Duffin–schaeffer‐type Conjectures in Various Local Fields

Abstract

Abstract. This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that λ(∩ ∞ m=1 ∪ ∞ n=m E n ) = 1 if and only if n λ(E n ) = ∞, where λ denotes the Lebesgue measure on R/Z,ψ is any non-negative arithmetical function. Instead of studying ∩ ∞ m=1 ∪∞ n=m E n we introduce an even fundamental object ∪ ∞ n=1 E n and conjecture there exists a universal constant C > 0 such thatIt is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena are found in the fields of p-adic numbers and formal Laurent series. As a byproduct, we answer conditionally a question of Haynes by showing that one can always use the quasi-independence on average method to deduce λ(∩ ∞ m=1 ∪ ∞ n=m E n ) = 1 as long as the Duffin-Schaeffer conjecture is true. We also show among several others that two conjectures of Haynes, Pollington and Velani are equivalent to the Duffin-Schaeffer conjecture, and introduce for the first time a weighted version of the second Borel-Cantelli lemma to the study of the Duffin-Schaeffer conjecture. Introduction to the Duffin-Schaeffer conjectureThroughout the paper we use the following notations:• p denotes a prime number, • n, h denote positive integers, • ϕ(n) denotes the Euler phi function, • λ denotes the Lebesgue measure on R/Z, r) (B(x, r)) denotes the open (closed) ball with center x and radius r in a given metric space.2000 Mathematics Subject Classification. 11J83.

Some Refined Results on the Mixed Littlewood Conjecture for Pseudo-Absolute Values

“…13 Zero-one law: λ(lim sup E n (ψ)) ∈ {0, 1} [12]. Subhomogeneity: For any t ≥ 1, λ(lim sup E n (tψ)) ≤ tλ(lim sup E n (ψ)) [17].…”

“…However Duffin-Schaeffer theorem requires good match between sequence ψ(n) and Euler function ϕ(n), so that hypotheses (5), ( 7) and ( 9) are very important. For some nice functions ψ(n), Duffin-Schaeffer theorem can be improved [2,[15][16][17]. We will use [17,Theorem 1.17] to study the mixed Littlewood conjecture and find that restriction ( 5) is not necessary in some sense.…”

“…For some nice functions ψ(n), Duffin-Schaeffer theorem can be improved [2,[15][16][17]. We will use [17,Theorem 1.17] to study the mixed Littlewood conjecture and find that restriction ( 5) is not necessary in some sense.…”

Badly approximable points for diagonal approximation in solenoids

“…There are several natural ways of extending the classical study of Diophantine approximation to spaces of p-adic numbers. The first and most obvious, which was pioneered by K. Mahler [24], Jarník [16], and Lutz [23], is to study approximations in Q p in the usual sense by simply restricting to Z p (see also [2,3,12,14,21,22,27]). A second way, which was introduced by Choi and Vaaler [7] and studied using techniques from the geometry of numbers over the adeles [4,5], is to work in projective space over Q p (see also [10,11]).…”