An effective proof of Roth's theorem on quadratic dispersion

Means over orbits of multidimensional lattices

“…Thus, it was proved that for the quadratic deviation the approaches in [5] and [6] give the same result. Moreover, in the paper [11] algorithmic problems of the search of a lattice from an orbit with quadratic deviation not exceeding the arithmetic mean over the orbit were considered.…”

“…Similarly, the Chen transformations [5] of Faure sets and the transformations from the paper [6] form finite groups with respect to composition.…”

“…Chen applied the Roth transformations to Hammersley lattices and introduced new discrete transformations for Faure sets. Dobrovoltskii in [6,7] proposed discrete transformations of Hammersley lattices; applying these transformations, he established Roth and Chen upper bounds with better constants. In a series of papers [7][8][9][10], algorithmic problems of the search of optimal transformations of Hammersley lattices were discussed.…”

“…S. Van'kova, N. M. Dobrovol'skii, and A. R. Esayan [11] singled out a class of rational /Y-lattices; for these lattices two transformation groups were defined, namely, the group of arithmetic shifts and the group of digital shifts. The group of arithmetic shifts is a generalization of the construction in [6,7] and the group of digital shifts is a generalization of the construction of Chen [5] for Faure sets.…”

“…In the papers [5][6][7] the existence of a lattice with a regular quadratic deviation or q-deviation was proved by estimating the mean q-deviation over an orbit of a Hammersley lattice or a Faure lattice.…”