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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.\n 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). 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