A result of Gallagher implies that for almost every $(\alpha,\beta)\in\mathbb{R}^2$
$$\liminf_{q\rightarrow\infty}\ q\log^2 q\ \|q\alpha \|\|q\beta \| = 0.$$
In the first part of the talk I will try to convince you that this result can be improved and thus we can expect more from Littlewood’s Conjecture from a metrical point of view. In the second part, I will investigate concrete situations in which inhomogeneous Diophantine approximation results can be derived from their homogeneous counterparts. For example, for any $i, j\ge0$ with $i + j = 1$ and $\gamma\in\mathbb{R}$, let $\mathbf{Bad}_\gamma (i, j)$ denote the set of points $(x, y)\in\mathbb{R}^2$ for which
$$\liminf_{q\rightarrow\infty} \ q \max\{\|qx − \gamma\|^{1/i} , \|qy − \gamma\|^{1/j}\} > 0.$$
Then the basic construction that proves the homogeneous result that $\dim \mathbf{Bad}_0 (i, j) =2$ can be naturally adapted to show that $\dim \mathbf{Bad}_\gamma (i, j) = 2$.