Strong row coresets: the even/non-even dichotomy

Let $S_p(k, \varepsilon)$ be the worst-case minimum support of a nonnegative
strong coreset on original rows for Euclidean subspace costs of power $p$,
with no restriction on input rank. For every fixed real $p > 2$ that is not an
even integer, the manuscript proves
$c_p k^{p/2}\varepsilon^{-2}\log^{-(5p/2+3)}(2k/\varepsilon) \le S_p(k, \varepsilon) \le C_p k^{p/2}\varepsilon^{-2}\log^{p+5}(2k/\varepsilon)$
for all $k \ge 1$ and $0 < \varepsilon < 1/2$; the upper bound is imported
from Lin, Mirrokni and Woodruff. The lower bound tensors a bounded-cost,
stably evaluable core with a selected Boolean-cube family, and the
nonvanishing middle Walsh spectrum of absolute non-even powers, a mechanism
with an antecedent in Li, Wang and Woodruff, supplies the accuracy factor.
A second part proves, for even $p$, the matching lower bound and, through a
structured Gaussian estimate for polynomial queries after a row reduction by
Lin, Mirrokni and Woodruff's bound, the upper bound
$\widetilde O_p(k^{(p+1)/2}\varepsilon^{-1} + k^{p/2-1}\varepsilon^{-2})$; with
their bound this gives
$S_p(k, \varepsilon) = \widetilde\Theta_p(\min\{k^{p/2}\varepsilon^{-2}, k^{(p+1)/2}\varepsilon^{-1} + k^{p/2-1}\varepsilon^{-2}\})$.
Hence the bound
$\widetilde O_p(k^{p/2}\varepsilon^{-1} + k\varepsilon^{-2})$ fails for every
fixed $p > 2$.