  Benford's law states that for many random variables X > 0 its leading digit D
= D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d
= 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact,
applied to Y := log_{10}(X): For many real random variables Y, the remainder U
:= Y - floor(Y) is approximately uniformly distributed on [0,1). The present
paper provides new explicit bounds for the latter approximation in terms of the
total variation of the density of Y or some derivative of it. These bounds are
an interesting alternative to traditional Fourier methods which yield mostly
qualitative results. As a by-product we obtain explicit bounds for the
approximation error in Benford's law.
