Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators
Abstract
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 \an\n∈N,\, \bn\n∈N,\, \cn\n∈N,\, \dn\n∈N, we determine convergence conditions under which the set of x∈ [0,1] which satisfy an x +cn · bn x + dn < (n) for infinitely many n∈N has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting.
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