Let \(A\) and \(B\) be two proper subsets of \(\mathbb{Z}_ n\) such that \(A+B\neq\mathbb{Z}_ n\). A theorem of Cauchy and Davenport states that \(A+B\geq| A|+| B|-1\) if \(n\) is a prime. Chowla generalized the Cauchy-Davenport theorem and proved that the same inequality holds if \(0\in B\) and \(\text{gcd}(x,n)=1\) for all nonzero \(x\in B\). The author proves in this paper that if \(0\in B\) and for all \(x,y\in B\backslash\{0\}\) such that \(x\neq y\) and \(\text{gcd}(x,y,n)=1\), then \(A+B\geq| A|+| B|-2\). Moreover, \(A+B\geq| A|+| B|-1\) unless \(| B|=2\) or there exists a \(b\in B\) such that \(B\backslash\{0,b\}\) is a union of cosets of modulo the cyclic generated by \(b\) in \(\mathbb{Z}_ n\). The proofs are based on some results on atoms in Cayley graphs proved earlier by the author some of which are listed in the paper.