We prove an inequality for characteristic functions of lattice distributions, which can be used to give "explicit" remainder term estimates in local central limit theorems.
"An Estimate of the Modulus of the Characteristic Function of a Lattice Distribution with Application to Remainder Term Estimates in Local Limit Theorems." Ann. Probab. 3 (1) 162 - 165, February, 1975. https://doi.org/10.1214/aop/1176996456