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 $(1+\text{vepsilon}{)}^n$ is at a distance greater than $2^{-17}ϵ\mathrm{\mid }\log \text{vepsilon}{\mathrm{\mid }}^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17}\mathrm{\mid }\log ϵ{\mathrm{\mid }}^{-1}$. 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.