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We give a thorough analysis of the gap between such estimates when $X=RP^m$, the real projective space of dimension $m.$ In particular, we describe a number $r(m)$, which depends on the structure of zeros and ones in the binary expansion of $m$, and with the property that $TC_s(RP^m)$ is given by $sm$ with an error of at most one provided $s \\geq r(m)$ and $m \\not\\equiv 3 \\bmod 4$ (the error vanishes for even $m$). 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