  We study the rate of growth of ergodic sums along a sequence (a_n) of times:
S_N f(x)=f(T^{a_1}x) + ... + f(T^{a_N}x). We characterize the maximal rate of
growth of these ergodic sums and identify a number of sequences such as (2^n)
that achieve this rate of growth.
  We also return to Khintchine's strong uniform distribution Conjecture which
stated that the averages (1/N)(f(x)+f(2x mod 1)+...+f(Nx mod 1)) converge
pointwise almost everywhere to \int f for an integrable function on [0,1). We
give an elementary counterexample to this conjecture, showing that divergence
occurs at the maximal rate. In addition, we consider versions of Khintchine's
conjecture in which the sequence of positive integers is replaced by the
increasing enumeration of a subsemigroup of the positive integers. We give
necessary and sufficient conditions on the semigroup for convergence in the
modified conjecture.
