On fractional parts of powers of real numbers close to 1

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