Primes in the form $[\alpha p+\beta ]$

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^(x)={p<=x: both p and [\alpha p+\beta] are primes}.