{"id":"palomar/PALOMAR-2026-09-02-000003","url":"https://github.com/stalex444/salem-theorem/tree/887be765bf24c6a645600dfc7bd0a1055a1761dd","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-02-000003-v1.json","repo":"https://github.com/stalex444/salem-theorem","tags":["math.NT","msc:11R06","msc:11R04","number-theory","salem-numbers","pisot-numbers","lean"],"score":0,"title":"Salem's theorem: every Pisot number is a two-sided limit of Salem numbers","trust":"high","domain":"math.NT","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.NT","summary":"Kernel-checked Salem's theorem — every Pisot number is a limit point of Salem numbers from both sides (R. Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath, 1963, Ch. III, Theorem IV, p. 30; origin R. Salem, Duke Math. J. 12 (1945)) — with both of Salem's constructions, each stated as a construction. Three compared statements carry the surface. (1) salem_theorem — the theorem for Pisot numbers: a real algebraic integer alpha > 1 every other complex root of whose minimal polynomial has modulus 0, a Salem number in (alpha - eps, alpha) and one in (alpha, alpha + eps). (2)…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-02","reactions":{},"formalized":true,"formalizes":[{"title":"Algebraic Numbers and Fourier Analysis","identifier":"Heath Mathematical Monographs, D. C. Heath and Company, Boston, 1963, x+68 pp. (reprinted Wadsworth, Belmont, 1983); Ch. III, Theorem IV, p. 30, proof pp. 30-31; the definition of the class T and its reciprocity consequence, p. 26 (cited by page; numbered Lemma 1 in Smyth 2015)","relationship":"formalizes"},{"title":"Power series with integral coefficients","identifier":"Duke Mathematical Journal 12 (1945), no. 1, 153-172","relationship":"formalizes"},{"title":"Pisot and Salem numbers in intervals of the real line","identifier":"Mathematics of Computation 32 (1978), no. 144, 1244-1260; Section 4, eq. (26), p. 1252","relationship":"adapts"}]}