On the convergence to uniform distribution

This paper considers sums of independent identically distributed random variables. We give an example in which, under the unbounded growth of a number of summands, the probability densities $\tilde{p}_n(x)$ of fractional parts of these sums converge to 1 in the sense of
$$ \int_{0}^1\bigl|\tilde{p}_n(x)-1\bigr|\,dx\to 0, $$
but they do not converge to 1 in the uniform metric
$$ \sup_{0\leq x\leq 1}\bigl|\tilde{p}_n(x)-1\bigr|. $$