= Solution
Interpolation between $L^2$ and $L^6$, followed by the three-dimensional <Sobolev inequality>, gives
$$
\|v\|_{L^3}
\leq\|v\|_{L^2}^{1/2}\|v\|_{L^6}^{1/2}
\leq c|v|^{1/2}\|v\|^{1/2}.
$$
Apply this with $v=P_Nw$ and use part i:
$$
\|P_Nw\|_{L^3}
\leq c|P_Nw|^{1/2}
\left(\lambda_N^{1/2}|P_Nw|\right)^{1/2}
=c\lambda_N^{1/4}|P_Nw|.
$$
Orthogonal projection is contractive in $H$, so
$$
\boxed{\|P_Nw\|_{L^3}
\leq c\lambda_N^{1/4}|w|}.
$$
The Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.
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