On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$

Sets of bounded discrepancy for multi-dimensional irrational rotation

We study bounded remainder sets with respect to an irrational rotation of the $d$-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one. First we extend to several dimensions the Hecke-Ostrowski result by constructing a class of $d$-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of "equidecomposability" to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.Comment: To appear in Geometric And Functional Analysi