On a variant of the large sieve
Abstract
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 <= N1/2, remove or ``sieve out'' by all n whose reduction mod p lies in some interval Ip of Z/pZ of length (p-1)/2. Let A be the set that remains: then |A| << N1/3 + o(1), a bound which improves slightly on the bound of |A| << N1/2 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 Lindel\"of hypothesis, Gauss circle problem and Dirichlet divisor problem) the bound can be improved to |A| << No(1). This raises the worry that even reasonably simple sieve problems are connected to issues of which we have little understanding at the present time.