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The $s$-th higher topological complexity of a space $X$, $TC_s(X)$, can beestimated from above by homotopical methods, and from below by homologicalmethods. We give a thorough analysis of the gap between such estimates when$X=RP^m$, the real projective space of dimension $m.$ In particular, wedescribe a number $r(m)$, which depends on the structure of zeros and ones inthe binary expansion of $m$, and with the property that $TC_s(RP^m)$ is givenby $sm$ with an error of at most one provided $s \geq r(m)$ and $m \not\equiv 3\bmod 4$ (the error vanishes for even $m$). The latter fact appears to beclosely related to the estimation of the Euclidean immersion dimension of$RP^m$. We illustrate the phenomenon in the case $m=3 \cdot 2^a$.

The stability of the higher topological complexity of real projective spaces: an approach to their immersion dimension