---

Source. Conjecture 1.2, p. 515, of Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics 164 (2006), 513--560, in the edition identified on the source card.

Statement

Conjecture 1.2 (Littlewood (c. 1930), p. 515; the paper cites Margulis's survey [24, §2] for it). Quoted from p. 515: "For every $u,v\in\mathbb R$,

$$\liminf_{n\to\infty} n\langle nu\rangle\langle nv\rangle=0, \qquad (1.1)$$

where $\langle w\rangle=\min_{n\in\mathbb Z}|w-n|$ is the distance of $w\in\mathbb R$ to the nearest integer."

The paper states (p. 515) that Margulis's Conjecture 1.1, recorded on the page of Theorem 1.3, implies Conjecture 1.2. It records three known partial facts on p. 516: (1.1) holds for almost every pair $(u,v)$, since already $\liminf_{n\to\infty}n\langle nu\rangle=0$ for almost every $u$; Cassels and Swinnerton-Dyer proved (1.1) for any $u,v$ from the same cubic number field; and Pollington and Velani proved that for every real $u$ the set of $v$ for which $(u,v)$ satisfies (1.1) meets the badly approximable numbers in a set of Hausdorff dimension one.

The paper does not prove the conjecture. Its result toward it is Theorem 1.5: the pairs violating (1.1) form a set of Hausdorff dimension zero.

Read depth. Claims checked: the statement and the surrounding remarks were read clause by clause on pp. 515--516.

Bears on

• Problem 495: the problem asks exactly whether (1.1) holds for all real $\alpha,\beta$, written with $\|x\|$ for the paper's $\langle x\rangle$. The paper poses it as an open conjecture and does not prove it.