Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Abstract

Let G be an abelian group, S ⊆ 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 ƒ : S → T to be ε-close on S to a character φ : G → T it is enough that ƒ be extendable to a mapping ¯f : ( S U {1} U S −1) n → T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary σ < min(ε, π/2). Moreover, n can be chosen uniformly, independently of G and both ƒ and ¯f, depending just on σ, 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 ∈ R q to be ε-close to some vector from the dual lattice H ★ of a full rank integral point lattice H ⊆ ℤ 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 Σ i| x i | < n, where n depends on δ, ε and q only.