Harmonious orbits of linear transformations (Q1912213)

A subset \(\Lambda\) of a locally compact abelian group \(G\) is said to be harmonious if any homomorphism of \(G\) into the circle group can be approximated uniformly on \(\Lambda\) by continuous ones [cf. \textit{Y. Meyer}'s book, Algebraic numbers and harmonic analysis, North Holland Math. Library, Vol. 2 (Amsterdam-London 1972; Zbl 0267.43001)], which serves as a general reference for the paper. A polynomial \(P\) with real coefficients and such that all its roots are algebraic integers is said to satisfy the condition \(PV\) if its roots satisfy a certain condition related to the notion of a Pisot-Vijayaraghavan number; a slightly weaker condition defines the condition \(PVS\) \((S\) refers to Salem numbers). Now let \(A\) be a nonsingular \(n \times n\) matrix, \(P\) its minimal polynomial, and let \(v_0 \in R^n\) be such that the orbit \(\Lambda = \{A^k v_0 \mid k \geq 0\}\) spans all of \(R^n\). Consider also the larger set \(\Lambda'\) consisting of all finite sums of different elements of \(\Lambda\). The following theorem is proved: the sets \(\Lambda\), \(\Lambda'\) are harmonious if and only if \(P\) satisfies \(PVS\), respectively \(PV\). Notice that the condition does not depend on the particular choice of \(v_0\). For the case \(n = 2\) such results were obtained by \textit{H. Ferru} [C. R. Acad. Sci., Paris, Sér. A 272, 658-660 (1971; Zbl 0207.12701); Bull. Soc. Math. France, Suppl., Mem. No. 19, 5-20 (1969; Zbl 0194.16401); Bull. Sci. Math., II. Ser. 92, 81-96 (1968; Zbl 0164.07401)]. In the present paper it is first proved that harmonicity implies the conditions \(PVS\), respectively \(PV\). The proof for the opposite implications is also given; here the arguments are said to be similar to the corresponding steps in Meyer's book.