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In his MathOverflow question "How thick is the reciprocal of the squares?" Kevin O'Bryant asks if a certain set, the reciprocal of the set of squares (identifying sets with power series in $F_2[[x]]$), has positive density. (If I've understood correctly the current situation is that the density is thought to be $\le 1/64$.) I was curious if anything is known about the comparable question for the reciprocal of the set of cubes.

Start with the series $1+x+x^8+x^{27}+...$ Take its reciprocal $1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^9-2x^{10}+3x^{11}... $ and consider the set of those $n$ for which $x^n$ has a coefficient which is odd. This sequence begins with: 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 16, 19, 20, 23, 29, 32, 34, 35, 37, 45, 47, 48, 49, 53, 54, 57, 58, 67, 69, 71, 73, 75, 85, 86, 99, 101,...

Examination of the first 25,000 shows that thus far the density seems to be about $.27$.

It looks to me that one might have a chance of categorizing those numbers in this sequence which are even. In particular it seems that numbers of the form $2k^3$, $4k^3$,... are there.

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    $\begingroup$ where does 1/64 come from? I thought it is known to be less than 1/16, and two available conjectures think it to be 0 or 1/32? or has something changed since 2010? $\endgroup$
    – Vladimir Dotsenko
    Commented Sep 11, 2014 at 6:47
  • $\begingroup$ @dotsenko: I believe that 1/64 is what Kevin O'Bryant told me in a conversation last week, but I could be misremembering. $\endgroup$
    – David S. Newman
    Commented Sep 12, 2014 at 1:41
  • $\begingroup$ It'd be interesting to have a confirmation of that if it is really the case. (My main references to date are arxiv.org/abs/1009.3985 and arxiv.org/abs/1107.4137). $\endgroup$
    – Vladimir Dotsenko
    Commented Sep 12, 2014 at 6:39
  • $\begingroup$ @VladimirDotsenko I think the person to ask is Paul Monsky. $\endgroup$
    – David S. Newman
    Commented Sep 20, 2014 at 1:18
  • $\begingroup$ It seems that the even numbers in the sequence mentioned are numbers which can be written in an odd number of ways as a sum 2a^3+4b^3, where a and b are non-negative numbers. $\endgroup$
    – David S. Newman
    Commented Oct 10, 2014 at 19:35

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