In Khinchin's book, "Continued Fractions," he considers the question, given an irrational, $\alpha$, and a real number, $\beta$, how to find integral $x$ and $y$ such that
$$\alpha x - y \approx \beta$$
to a given level of accuracy.


He then says that Chebyshev "obtained the first basic results connected with it, and has been the subject of continued intensive study, especially by the Soviet arithmetic school." Khinchin goes on to prove a few interesting theorems concerning this question, but then moves on to other topics.


Where can I find more about work that has been done on this problem?



Continued fractions, Chebyshev and non-homogenous approximation
Our question is a very special case of the general problem of restricted simultaneous Diophantine approximation. Thus, I believe that the best place to start is to consult the paper of Schmidt: *Two questions in Diophantine approximation*, Monatsh. Math. 82, 237-245. Moreover, you also should consult the paper of P. Thurnheer: *On Dirichlet's theorem concerning diophantine approximation*, Acta Arithmetica 54 (1990), 241--250 and reference therein.


