Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\, \{b_n\}_{n\in\mathbb{N}},\, \{c_n\}_{n\in\mathbb{N}},\, \{d_n\}_{n\in\mathbb{N}},$ we determine convergence conditions under which the set of $x\in [0,1]$ which satisfy $\left\lVert a_n x +c_n\right\rVert \cdot \left\lVert b_n x + d_n \right\rVert < \psi(n)$ for infinitely many $n\in\mathbb{N}$ has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting.