  Chinese Remainder Theorem (CRT) has been widely studied with its applications
in frequency estimation, phase unwrapping, coding theory and distributed data
storage. Since traditional CRT is greatly sensitive to the errors in residues
due to noises, the problem of robustly reconstructing integers via the
erroneous residues has been intensively studied in the literature. In order to
robustly reconstruct integers, there are two kinds of traditional methods: the
one is to introduce common divisors in the moduli and the other is to directly
decrease the dynamic range. In this paper, we take further insight into the
geometry property of the linear space associated with CRT. Echoing both ways to
introduce redundancy, we propose a pseudo metric to analyze the trade-off
between the error bound and the dynamic range for robust CRT in general.
Furthermore, we present the first robust CRT for multiple numbers to solve the
problem of the CRT-based undersampling frequency estimation in general cases.
Based on symmetric polynomials, we proved that in most cases, the problem can
be solved in polynomial time efficiently. The work in this paper is towards a
complete theoretical solution to the open problem over 20 years.
