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We study the local limit theorem for weighted sums of Bernoulli variables. Weshow on examples that this is an important question in the general theory ofthe local limit theorem, and which turns up to be not well explored. Theexamples we consider arise from standard random models used in arithmeticalnumber theory. We next use the characteristic function method to prove newlocal limit theorems for weighted sums of Bernoulli variables. Further, we givean application of the almost sure local limit theorem to a representationproblem in additive number theory due to Burr, using an appropriate randommodel. We also give a simple example showing that the local limit theorem, inits standard form, fails to be sharp enough for estimating the probability$P\{S_n\in E\}$ for infinite sets of integers $E$, already in the simple casewhere $S_n$ is a sum of $n$ independent standard Bernoulli random variables and$E$ an arithmetic progression.

Local Limit Theorems in some Random models from Number Theory