String distance communication complexity
After the OP's edit (see comments below), this answer is outdated and does not address the question. Leaving it for the said comments.



This seems to be asking the same as in this question of mine (which only has a partial answer, fitting for a partial function). Quoting from the question:


  [Consider, for $x,y\in\{0,1\}^n$] the partial function
  $$
\textsf{GHD}_{n,t,g} = \begin{cases}
0 &amp; \text{ if }\operatorname{d}_H(x,y) \leq t-g\\
1 &amp; \text{ if }\operatorname{d}_H(x,y) \geq t+g.
\end{cases}$$
  Lemma 4.1 to Proposition 4.4 of [CR10] allow to get a lower bound on the communication complexity of $\textsf{GHD}_{n,t,g}$ for (most) of the settings of $t,g$. 


As answered there, the general one-sided version is solved in the following paper of Egor Klenin and Alexander Kozachinsky [1], who show that (with your notations) $\tilde{\Theta}\!\left(\frac{t}{(1-\alpha)^2}\right)$ bits are necessary and sufficient.

[1] One-sided error communication complexity of Gap Hamming Distance, Egor Klenin and Alexander Kozachinsky, 2016. ECCC TR16-173.