Rational Spectra for Finite Hadamard Pairs and Bounded Spectral Sets on the real line

We prove that every finite spectral pair \((A,\Gamma)\) with \(A\subset\mathbb Z\), \(\Gamma\subset\mathbb R/\mathbb Z\), and \(0\in\Gamma\) has a rational spectrum, that is, \(\Gamma\subset\mathbb Q/\mathbb Z\).

Our argument relies on a modulus rigidity theorem for generalizedVandermonde systems satisfying inverse-orthogonality relations, constructed by hyperbolic positive-definite kernels.

Combined with Galois conjugation and Kronecker's theorem, this rigidity forces the associated exponential nodes to be roots of unity.

As an application, using the periodicity and fiberization of one-dimensional spectra, we show that every spectrum \(\Lambda\) of a bounded measurable spectral set \(\Omega\subset\mathbb R\) with \(|\Omega|=1\) and \(0\in\Lambda\) is contained in \(\mathbb Q\).

This rational-spectrum result completes a chain of equivalences proven by Dutkay and Lai, which reduces the full one-dimensional Fuglede's conjecture to its finite cyclic analogues over \(\mathbb{Z}_n\).