Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9521148 | Indagationes Mathematicae | 2005 | 14 Pages |
Abstract
Let G be an abelian group, S â G be a finite set, and T denote the multiplicative group of complex unitswith the invariant arc metric | arg(a/b)|. We will show that for a mapping Æ : S â T to be ε-close on S to a character Ï : G â T it is enough that Æ be extendable to a mapping ¯f : (S U {1} USâ1)n â T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary Ï < min(ε, Ï/2). Moreover, n can be chosen uniformly, independently of G and both Æ and ¯f, depending just on Ï, ε and the number of elements of S. The proof is non-constructive, using the ultraproduct construction and Pontryagin duality, hence yielding no estimate on the actual size of n. As one of the applications we show that, for a vector u â R q to be ε-close to some vector from the dual lattice Hâ
of a full rank integral point lattice H â â¤q, it is enough for the scalar product ux to be δ-close (with δ < 1/3) to an integer for all vectors xH satisfying Σi|xi | < n, where n depends on δ, ε and q only.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Martin MaÄaj, Pavol ZlatoÅ¡,