Let $\alpha$ be a real irrational algebraic number. The now-classic Thue-Siegel-Roth theorem asserts that for any $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \varepsilon)$ such that for all rational numbers $p/q$ with $\gcd(p,q) = 1, q > 0$ we have
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert > \frac{c}{q^{2 + \varepsilon}}.$$
Equivalently, this says that for the linear form $x - \alpha y$ one has
$$\displaystyle \left \lvert x - \alpha y \right \rvert > c |y|^{-1 - \varepsilon}$$
provided that $x,y \in \mathbb{Z}, y \ne 0$.
My question is whether something similar can be said for other ranges. Fix $0 < \theta < 1$ (independent of $\varepsilon$) and $\beta$ another real algebraic number. Can one find infinitely many $x,y \in \mathbb{Z}$ such that
$$\displaystyle |x - \alpha y - \beta y^\theta| = o(1)?$$
Further, can one obtain a Thue-Siegel-Roth type result? That is, does there exist a $\psi = \psi(\theta) > 0$ such that for all $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \beta, \theta, \varepsilon)$ such that
$$\displaystyle |x - \alpha - \beta y^\theta| > c q^{-\psi - \varepsilon}$$
for all $x,y \in \mathbb{Z}$ with $x,y \ne 0$?
