On a Variant of the Large Sieve

We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N ] = {1, . . . , N} and, for each odd prime p 6 √ N , remove or “sieve out” by all n whose reduction n(mod p) lies in some interval Ip ⊆ Z/pZ of length (p−1)/2. Let A be the set that remains: then |A| ≪ N log N , a bound which improves slightly on the bound of |A| ≪ N logN which results from applying the large sieve in its usual form. This is a very, very weak result in the direction of a question of Helfgott and Venkatesh, who suggested that nothing like equality can occur in applications of the large sieve unless the unsieved set is essentially the set of values of a polynomial (e.g. A is the set of squares). Assuming the “exponent pairs conjecture” (which is deep, as it implies a host of classical questions including the Lindelöf hypothesis, Gauss circle problem and Dirichlet divisor problem) we can improve the bound to |A| ≪ N. This raises the worry that even reasonably simple sieve problems are connected to issues of which we have little understanding at the present time.