Given two coprime integers $a < b$ of different parities, only a finite number of points in $\mathbb N^2$ cannot be reached by a walk in $\mathbb N^2$, starting at the origin and using only steps of the form $(b,\pm a),(\pm a,b)$ (and thus making an acute angle with the north-eastern vector $(1,1)$).

Is there a good upper bound on the number of such exceptional points? Is there a good upper bound on the coordinate sum $x+y$ of such an exceptional point $(x,y)$?

(Remark: A naive proof that almost all points can be reached gives an upper bound which is probably very far from the true value.)