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 \textit{ homothetic quadratic discrepancy} \begin{ equation*} \mathcal{ D}_2(\mathcal{ P},\, C)=\int_{ 0}^{ 1}\int_{ [0,1)^2}\left|\mathcal{ D}( \mathcal{ P},\,\boldsymbol{ \tau}+\delta C)\right|^2\,{ \rm d}\boldsymbol{ \tau}\,{ \rm d} \delta. \end{ equation*} We investigate the behaviour of the optimal \textit{ homothetic quadratic 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 \textit{ 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 \textit{ h.q.d.} of planar convex bodies with a $\mathcal{ C}^2$ boundary has an order of growth of $N^{ 1/2}$. We show that, in general, a single order of growth for the optimal \textit{ h.q.d.} need not exist. First, by an implicit geometric construction of $C$, we obtain prescribed oscillations between $\log N$ and $N^{ 1/2}$. Second, by a subtler design of $\partial C$ and via Fourier-analytic methods, we obtain prescribed polynomial-order oscillations in the range $N^\alpha$ with $\alpha\in(2/5,1/2)$.
Categories: math.CA math.NT