Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

Abstract

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis.

Refinable Functions with PV Dilations

“…The first assertion follows from (3.2). Since S(A) is an real-analytic set it is homeomorphic to a union of embedded manifolds by Lojasiewicz's structure theorem for real-analytic sets [17], [21], [24]. Since θ(R) is a uniformly distributed embedding, if the dimension of S(A) were less than d − 1 then d(S( a)) = 0.…”

“…Lagarias and Yang conjectured [18] that certain real-analytic subsets of T n , that arise in the construction of refinable functions of several variables related to tilings and that are analogous to our set S(A), are simple. We used Lojasiewicz's theorem [21] to prove their conjecture. Thus we find the following result interesting:…”

Multiresolution Analyses on Quasilattices

“…Remark 3 Since the zero set of P trig is a real analytic set, an alternative proof based on Lojasiewicz's structure theorem [20], [27] for real analytic sets may be possible using methods that we developed in [26] to prove the Lagarias-Wang Conjecture.…”

“…Remark 8 Lemmas 15 and 16 ensure that V (P trig ) contains a rich set of points whenever f is integrable. We think that the techniques that we used to prove the Lagarias-Wang conjecture in [26], suitably modified, can be applied to prove Conjecture 2 by showing that there do not exist proper real algebraic subsets of T n that contains these rich set of points.…”

Filters and Functions in Multi-scale Constructions: Extended Abstract

“…Remark Since the zero set of P trig is a real analytic set, an alternative proof based on Lojasiewicz's structure theorem [16], [22] for real analytic sets may be possible using methods that we developed in [21] to prove the Lagarias-Wang Conjecture.…”