  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 <=
N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some
interval I_p of Z/pZ of length (p-1)/2. Let A be the set that remains: then |A|
<< N^{1/3 + o(1)}, a bound which improves slightly on the bound of |A| <<
N^{1/2} 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| <<
N^{o(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.
