Reformulating the p-adic Littlewood Conjecture in terms of infinite loops mod p
Reformulating the p-adic Littlewood Conjecture in terms of infinite loops mod p
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用无限循环 mod p 重新表述 p 进利特尔伍德猜想
DOI：
10.1016/j.jnt.2023.02.008
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发表时间：
2023
影响因子：
0.7
通讯作者：
Blackman J
中科院分区：
文献类型：
--
作者：
Blackman J
本文介绍了无限循环 mod n 的概念并讨论了它们的性质。特别是，它描述了无限循环的连分式展开式在乘以整数 n 时表现不佳。无限循环本质上是几何的，源于将连续分数视为双曲平面中的切割序列，但是，它们在丢番图近似方面也有一个很好的描述：无限循环 mod n 是任何不具有可被 n 整除的半收敛的实数。本文的主要结果是用无限循环重新表述 p 进利特尔伍德猜想 (pLC)。更明确地说，本文表明实数 α 是 pLC 的反例，当且仅当存在某个 m ∈ N 使得 p ℓ α 是一个无限循环 mod p m，对于所有 ℓ ∈ N∪{0}。
This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p-adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m∈ N such that p ℓ α is an infinite loop mod p m, for all ℓ∈ N∪{0}.