For non-integer j > 0 there exist signs making partial sums of k^j dense in R, via Thue-Morse blocks whose sums tend to zero with divergent variation plus a steering argument.
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Dense signed sums of non-integer powers
For non-integer j > 0 there exist signs making partial sums of k^j dense in R, via Thue-Morse blocks whose sums tend to zero with divergent variation plus a steering argument.