Polynomial-order oscillations in geometric discrepancy

Let $C\subset {\mathbb{R}}^2$ be a convex body, and for a positive integer $N$, let $\mathcal{P}$ be a configuration of $N$ points in $[0,1{)}^2$. The discrepancy of $\mathcal{P}$ with respect to $C$ is defined by \begin{equation*} $\mathcal{D}$($\mathcal{P}$,, C)=\sum_{$\mathbf{p}$\in$\mathcal{P}$}\sum_{$\mathbf{n}$\in$\mathbb{Z}$^2}$\mathbf{1}$C($\mathbf{p}$+$\mathbf{n}$)-N|C|, \end{equation*} and one may estimate how $\mathcal{P}$ deviates from uniformity by averaging the latter quantity over a family of sets. When considering quadratic averages over translated and dilated copies of $C$, one gets the $\mathit{\text{homothetic}}\mathit{\text{quadratic}}\mathit{\text{discrepancy}}$ \begin{equation*} $\mathcal{D}$2($\mathcal{P}$,, C)=\int{0}^{1}\int{[0,1)^2}\left|$\mathcal{D}$( $\mathcal{P}$,,\boldsymbolτ+δC)\right|^2,{\rm d}\boldsymbolτ,{\rm d} δ. \end{equation*} We investigate the behaviour of the optimal $\mathit{\text{homothetic}}\mathit{\text{quadratic}}\mathit{\text{discrepancy}}$, that is \begin{equation*} \inf_{# $\mathcal{P}$=N} $\mathcal{D}$_2($\mathcal{P}$,, C)\quad$\text{as}$\quad N\to+\infty. \end{equation*} Beck~\cite{MR915529} and Beck and Chen~\cite{MR1489133} showed that the optimal $\mathit{\text{h.q.d.}}$ of convex polygons has an order of growth of $\log N$, and more recently, Brandolini and Travaglini~\cite{MR4358540} proved that the optimal $\mathit{\text{h.q.d.}}$ of planar convex bodies with a ${\mathcal{C}}^2$ boundary has an order of growth of $N^{1\mathrm{/}2}$. We show that, in general, a single order of growth for the optimal $\mathit{\text{h.q.d.}}$ need not exist. First, by an implicit geometric construction of $C$, we obtain prescribed oscillations between $\log N$ and $N^{1\mathrm{/}2}$. Second, by a subtler design of $\mathrm{\partial }C$ and via Fourier-analytic methods, we obtain prescribed polynomial-order oscillations in the range $N^{\alpha }$ with $\alpha \in (2\mathrm{/}5,1\mathrm{/}2)$.