On fractional parts of powers of real numbers close to 1

الأصل · EN

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