The Fekete – Szegő theorem with splitting conditions : Part

A classical theorem of Fekete and Szegő [4] says that if E is a compact set in the complex plane, stable under complex conjugation and having logarithmic capacity γ(E) ≥ 1, then every neighborhood of E contains infinitely many conjugate sets of algebraic integers. Raphael Robinson [5] refined this, showing that if E is contained in the real line, then every neighborhood of E contains infinitely many conjugate sets of totally real algebraic integers. In [2], David Cantor developed a theory of capacity for adelic sets in P1. One of his key results was a very strong theorem of Fekete–Szegő–Robinson type, which produced algebraic numbers whose conjugates lay in a specified neighborhood of an adelic set E = E∞× ∏ pEp, and belonged to P1(R), and P(Qp) for finitely many primes p (“splitting conditions”). Unfortunately there was a gap in the part of the proof concerning the splitting conditions. Some time ago the author extended Cantor’s theory, including the Fekete–Szegő theorem without splitting conditions, to arbitrary algebraic curves [6]. This paper represents a step towards establishing the theorem with splitting conditions. We prove the theorem in the special case where the ground field is Q, the sets are E∞ = [−2r, 2r] and Ep = Zp for primes p in a finite set T , and capacities are measured relative to the point ∞. It will be apparent to anyone familiar with this kind of result that we have drawn ideas from earlier papers. The method of proof, called “patching”, goes back to Fekete and Szegő [4]. The use of Chebyshev polynomials for the archimedean patching functions comes from Robinson [5], and the use of Stirling polynomials for the p-adic patching functions comes from Cantor [2]. However, we have introduced several new ideas: in particular, the method for preserving “well-distributed” sequences of roots of p-adic polynomials, and