  We consider the large sieve inequality for sparse sequences of moduli and
give a general result depending on the additive energy (both symmetric and
asymmetric) of the sequence of moduli. For example, in the case of monomials
$f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the
previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018)
and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in
\mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We
then apply our results to obtain a version of the Bombieri--Vinogradov theorem
with Piatetski-Shapiro moduli improving the level of distribution of
R.~C.~Baker (2014).
