Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.

Various $p$-adic versions of Littlewood's conjecture are investigated, generalizing a set-up considered recently by de Mathan and Teulié. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to $\xi$ and $\xi^2$ by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers $\xi$. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3.