Toward Khintchine's theorem with a moving target: Extra divergence or finitely centered target

Abstract Szüsz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on under which for almost every real number there exist infinitely many rationals such that where is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of to a shrinking target centered around , viewed in . Hauke and the second author have conjectured that Szüsz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter is allowed to depend on the denominator of the approximating rational. We show that the conjecture holds under an “extra divergence” assumption on . We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite‐colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.