Primes of the form ⌊αp+β⌋⌊αp+β⌋

Let β   be a real number. Then for almost all irrational α>0α>0 (in the sense of Lebesgue measure),lim supx→∞πα,β∗(x)(logx)2/x⩾1, whereπα,β∗(x)={p⩽x:both p and ⌊αp+β⌋ are primes}.