  We consider the Cauchy problem of the nonlinear Schr\"odinger equation with
the modulated dispersion and power type nonlinearities in any spatial
dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K.
Chouk and M, Gubinelli, Comm. Partial Differential Equations 40 (2015)] and
multilinear estimates which are based on divisor counting, and show the local
well-posedness. Our result generalizes the result by Chouk-Gubinelli.
