On asymptotics of large deviation probabilities of sums $\Sigma f(x2^n)$

In the paper, large deviations in limit theorems for sums $\sum_nf(x2^n)$ where $f(t)$ satisfies a Lipschitz condition, are investigated. A result is obtained for the narrow zone $[0,o((N/\ln^2N)^{1/4})]$. For the particular case $f(t)=\{t\}$ by a special method, a result for $[0,o(\sqrt N)]$ is obtained.