Counting multiplicative approximations

Abstract

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (Chin Ann Math 2:1–12, 1981) established an asymptotic formula for the number of such approximations, valid almost always. Using the quantitative Koukoulopoulos–Maynard theorem of Aistleitner–Borda–Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.

Appendix A: Computing a volume

Here, we deduce Theorem 1.6 from [9, Theorem 4.6] and the argument of Wang and Yu [13, Section 1]. For ${\lambda }> 0$, define

$$\begin{aligned} \mathcal B_k({\lambda }) = \{ \mathbf{x}\in [0,1]^k: 0 \le x_1 \cdots x_k \le {\lambda }\} \end{aligned}$$

and

$$\begin{aligned} \mathcal C_k({\lambda }) = \{ \mathbf{x}\in [0,1/2]^k: 0 \le x_1 \cdots x_k \le {\lambda }\}. \end{aligned}$$

By symmetry and [9, Theorem 4.6], for almost all ${\varvec{{\alpha }}}\in \mathbb R^k$, we have

$$\begin{aligned} S_\mathbf{0}^\times ({\varvec{{\alpha }}}, N, \psi ) = T_k(N) + O(\sqrt{T_k(N)} (\log T_k(N))^{2+\varepsilon }), \end{aligned}$$

where

$$\begin{aligned} T_k(N) = 2^k \sum _{n \le N} \mu _k(\mathcal C_k(\psi (n))). \end{aligned}$$

Thus, it remains to show that

$$\begin{aligned} T_k(N) \sim \Psi _k^\times (N) \qquad (N \rightarrow \infty ). \end{aligned}$$

(A.1)

Lemma A.1

For $k \in \mathbb N$ and ${\lambda }> 0$, we have

$$\begin{aligned} \mu _k(\mathcal B_k({\lambda })) = {\left\{ \begin{array}{ll} 1, &{}\text {if } {\lambda }\ge 1, \\ {\lambda }\sum _{s=0}^{k-1} \frac{(-\log {\lambda })^s}{s!}, &{}\text {if } 0< {\lambda }< 1. \end{array}\right. } \end{aligned}$$

Proof

We induct on k. The base case is clear: $\mu _1(\mathcal B_1({\lambda })) = \min \{ {\lambda }, 1\}$. Now let $k \ge 2$, and suppose the conclusion holds with $k-1$ in place of k. We may suppose that $0< {\lambda }< 1$. We compute that

$$\begin{aligned} \mu _k(\mathcal B_k({\lambda }))&= \int _0^1 \mu _{k-1} (\mathcal B_{k-1}({\lambda }/x)) {\partial }x = {\lambda }+ \int _{\lambda }^1 \frac{{\lambda }}{x} \sum _{s = 0}^{k-2} \frac{(\log (x/{\lambda }))^s}{s!} {\partial }x \\&= {\lambda }+ \sum _{s=0}^{k-2} \frac{{\lambda }}{s!} \int _1^{1/{\lambda }} \frac{(\log y)^s}{y} {\partial }y = {\lambda }+ {\lambda }\sum _{s=0}^{k-2} \frac{ (- \log {\lambda })^{s+1}}{(s+1)!} \\&= {\lambda }\sum _{t=0}^{k-1} \frac{(-\log {\lambda })^t}{t!}. \end{aligned}$$

$\square$

In view of Schmidt’s Theorem 1.4, we may assume that $k \ge 2$. Now, as

$$\begin{aligned} \mu _k(\mathcal C_k({\lambda })) = 2^{-k} \mu _k(\mathcal B_k(2^k {\lambda })), \end{aligned}$$

and as $\psi (n) < 2^{-k}$ for large n, we have

$$\begin{aligned} T_k(N)&= O_{k,\psi }(1) + \sum _{n\le N} \psi (n) \sum _{s=0}^{k-1} \frac{(- \log (2^k \psi (n)))^s}{s!} \\&= \Psi _k^\times (N) + O_k(\Psi _{k-1}^\times (N)) + O_{k,\psi }(1). \end{aligned}$$

Since $\psi (n) \rightarrow 0$ as $n \rightarrow \infty$, we have $\Psi _{k-1}^\times (N) + 1 = o(\Psi _k^\times (N))$, and hence, (A.1), completing the proof of Theorem 1.6.