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We study the connection between the singularities of a finite type$\mathbb{Z}$-scheme X and the asymptotic point count of X over various finiterings. In particular, if the generic fiber$X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is alocal complete intersection, we show that the boundedness of$\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ isreduced and has rational singularities. This paper completes a result ofAizenbud and Avni.

On rational singularities and counting points of schemes over finite rings