Andreas Weingartner <weingartner@suu.edu> asks:

<< Prove or disprove: There exists an epsilon > 0, such that no natural number has the property that in base 2 as well as in base 3, [a fraction of at most epsilon] of the digits are nonzero.

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At 07:58 PM 12/21/02 -0500, asimovd@aol.com wrote:

I hereby conjecture that for any epsilon > 0 there is some 2^K and some 3^L such that either of them, expressed in either base 2 or base 3, has a fraction < delta of digits that are nonzero.

[My intuition for this is partly based on the yet unproven -- but strongly supported by numerical evidence -- conjecture that the fractional parts of (3/2)^n, n = 1,2,3,..., are uniformly distributed in [0,1).]

I see what Dan is driving at, yes I do. But I'm not sure I share the clear intuition. Here are my difficulties. Dividing 3^n by 2^n does not "submerge" all of its bits, just the final n of them. The leading n lg 3 bits are left free to misbehave without disturbing the uniform distribution of the rest of the bits. Also, this seems to depend on the divergence of the sum of (n choose [n*epsilon]) 2^n. Gosper or Dan Asimov may have an intuition about whether this sum diverges. Maybe I could work it out if I sweated a bit, but intuition ... nope. Looking forward to enlightenment, I remain, yours sincerely, etc etc ... -A