A biconservative hypersurface M in a space form M^n+1(c) is a hypersurface whose normal part of the vector ΔH⃗ vanishes. We study such hypersurfaces with harmonic curvature or parallel Ricci curvature, and prove that have constant mean curvature. Also, by combining our results with previous results of H.B. Lawson Jr. and M. Umehara we obtain that if M is a biconservative hypersurface in a space form M^n+1(c) with parallel Ricci tensor or harmonic curvature, then the shape operator A of M is covariant constant, i.e. ∇A=0.
Let b ≥ 2 be an integer and ξ an irrational real number. We establishes that, if the irrationality exponent of ξ is less than 2.324 …, then the b-ary expansion of ξ cannot be `too simple', in a suitable sense. This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017].
