  By using a variant of Kowalski's large sieve for Frobenius in compatible
systems, we obtain zero-density estimates for arguments of $\ell$-adic trace
functions over finite fields with values in some algebraic subsets of the
cyclotomic integers, when the monodromy groups are known. This applies in
particular to hyper-Kloosterman sums and general exponential sums considered by
Katz.
