Defining ideals of some numerical semigroup rings with arithmetic pseudo-Frobenius numbers

In this paper, we study defining ideals of numerical semigroup rings. Let $H$ be a numerical semigroup with multiplicity $a_0$ and embedding dimension $n$. Assuming $a_0/2+1\leq n$, we prove that the defining ideal of $H$ is determinantal when the set of pseudo-Frobenius numbers forms an arithmetic sequence of length $n-1$. This partly resolves a conjecture of Cuong, Kien, Truong and, Matsuoka.