On fractional parts of powers of real numbers close to 1

We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} \epsilon |\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log \epsilon|^{-1}$. We also establish that the set of real numbers $\alpha > 1$ such that the sequence of fractional parts $(\{\alpha^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.