  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.
