In Harman's book "Prime Detecting Sieves" he describes a method to prove that a set contains primes if we have enough Type I and Type II information for it. As shown by Selberg's example of the set of numbers with an even number of prime factors Type I information is not sufficient to detect primes. Harman's methods require the Type II information to be given on sufficiently long intervals so that the sums we cannot give assymptotic formulae for are sufficiently small. Is it possible to prove, by giving a suitable counterexample, that if we have Type I information but only a very small amount of Type II information then we can't detect primes?

Let $A$ be the set in which we are interested, and suppose that $A\subseteq B$, the set of integers in $[x/2,x)$. Then, for a suitably small error $E$ and arbitrary bounded coefficients $a_m,b_n$ a Type I estimate is of the form $\sum_{mn\in A,m\leq M}a_m=\lambda \sum_{mn\in B,m\leq M}a_m+O(E)$ and a Type II sum is of the form $\sum_{mn\in A,x^\alpha\leq m\leq x^{\lpha+\beta}}a_mb_n=\lambda \sum_{mn\in B,x^\alpha\leq m\leq x^{\lpha+\beta}}a_mb_n+O(e).$ I am then wondering whether there are examples of sets satisfying these conditions for some $M,\alpha$ and sufficiently small $\beta$ which contain no primes. For definiteness take $M=x^{3/4}$ and $\alpha=1/4$.