Expected Discrepancy of Generalized Lattice Point Sets

Definitions and mathematical statement

Let $d,n\in{\mathbb{N}}$ and $p\in\{2,4,6,\ldots\}$. Choose $Z,\Delta$ independently and uniformly from $[0,1)^d$, and set \[ X_j=\{jZ+\Delta\}\quad(0\leq j<n),\qquad M_n^{Z,\Delta}=(X_0,\ldots,X_{n-1}), \] where braces denote coordinatewise fractional part. Separately, let $U_0,\ldots,U_{n-1}$ be independent uniform points in $[0,1)^d$, and put $M_n=(U_0,\ldots,U_{n-1})$. This notation treats both configurations as $n$ sampled points, with multiplicity; coincidences have probability zero for these distributions.

The source uses $\operatorname{disc}_p$ without reproducing its definition. Under the usual anchored-box $L^p$ convention, for $P=(x_0,\ldots,x_{n-1})$ define \[ D_P(t)=\frac1n\sum_{j=0}^{n-1}\mathbf1_{[0,t_1)\times\cdots\times[0,t_d)}(x_j)-\prod_{r=1}^d t_r, \qquad \operatorname{disc}_p(P)^p=\int_{[0,1)^d}|D_P(t)|^p\,{\,\mathrm{d}} t. \] On this convention, the cleaned source question is whether, for every $d,n\in{\mathbb{N}}$ and every even positive integer $p\geq2$, \[ {\mathbb{E}}_{Z,\Delta}\left[\operatorname{disc}_p(M_n^{Z,\Delta})^p\right] \leq {\mathbb{E}}\left[\operatorname{disc}_p(M_n)^p\right]. \] The anchored-box convention must be verified in the report; alternative discrepancy definitions are not automatically equivalent. Replacing the normalized counting error by its unnormalized version multiplies both sides by $n^p$, provided the same convention is used on both sides.

The original short prose speaks of expected $L^p$ discrepancy, while the cleaned, reportedly source-verified statement explicitly compares expectations of its $p$-th power. The latter is retained here: an inequality for the expected norm is a different assertion and must not be substituted. The supplied literature review checked 2026-08-21 reports that the checked shifted-lattice literature did not prove this displayed comparison and labels it open.

Short English statement

For even $p$, compare the expected $p$-th power of discrepancy for one uniformly randomized arithmetic progression on the torus with that for $n$ independent uniform points. The exact discrepancy convention still needs confirmation from the report.

Sources

• Local UnsolvedMath registry, permanent ID 30000052, catalogue entry, Expected Discrepancy of Generalized Lattice Point Sets.
• Discrepancy Theory and Its Applications (2004), Oberwolfach Reports, pp. 673--722, DOI https://doi.org/10.4171/owr/2004/13. The literature field also supplies https://doi.org/10.4171/OWR/2004/13.
• Supplied reference labeled "Shifted lattice rules": https://doi.org/10.1016/j.cam.2009.06.002.
• Supplied statement-verification field reports checking the official report; supplied literature triage is dated 2026-08-21. Neither check is independently repeated here.