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Cycle lengths modulo $k$ in expanders

Given a constant $α>0$, an $n$-vertex graph is called an $α$-expander if every set $X$ of at most $n/2$ vertices in $G$ has an external neighborhood of size at least $α|X|$. Addressing a question posed by Friedman and Krivelevich in [Combinatorica, 41(1), (2021), pp. 53--74], we prove the following result: Let $k>1$ be an integer with smallest prime divisor $p$. Then for $α>\frac{1}{p-1}$ every sufficiently large $α$-expanding graph contains cycles of length congruent to any given residue modulo $k$. This result is almost best possible, in the following sense: There exists an absolute constant $c>0$ such that for every integer $k$ with smallest prime divisor $p$ and for every positive $α<\frac{c}{p-1}$, there exist arbitrarily large $α$-expanding graphs with no cycles of length $r$ modulo $k$, for some $r \in \{0,\ldots,k-1\}$.