English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We show that the Lagrange spectrum L(Q [^/D]} of a real quadratic field has limit points. In the process we prove a curious relationship between the periods of the continued fractions of certain quadratic numbers. R6suM£. Nous demontrons que Ie spectre de Lagrange, L{Q[^/D]), d'un corps quadratique reel a des points d*accumulation. En passant, nous obtenons un rapport entre les periodes des fractions continues de certains nombres quadratiques. Let D be a non-square positive integer. Cusick and Mendes-France [1] have asked whether the Lagrange spectrum of Q[^/D] contains limit points. Woods [2] (in examining the equivalent problem for the Markov spectrum) has shown that the answer is yes in the case D=5. The main purpose of this paper is to show that the general answer is yes. To do this we establish an interesting Theorem on continued fractions which, more or less, generalizes the result used by Woods. I should like to acknowlege the helpful conversations that I have had with Mendes-France. We give a short resume of the facts we need. If aeR then its Lagrange number, L(a), is defined to be Um sup (q || a q ||)~". Here || , || denotes "distance from the nearest integer". L(a) is a measure of the approximability of a by rational numbers. If aeQ[^/D] then L(a)=2 lim sup^oo Irr(aJ where the a, are the successive remainders in the development of a as a continued fraction and Irr a = "irrational part of a". Hence, if [l» ' ' 9 t9H • • • » s}9 L(a)=2Max^ Irr[m^. . .w,Wi. . .m^-i] (*) This work was done while the author, in receipt of a Fellowship from the (British) Science Research Council, was enjoying the hospitality of the University of Bordeaux. Texte rccu Ie 19 mars 1979. revise Ie 26 juin 1979. S. WILSON, Department of Mathematics, Science Laboratories, South Road, Durham, DH13LE,Angleterre. BULLCT1N DE LA SOOETE MATHfeMATlQUE DE FRANCE 0037-9484/1980/137/S 5.00 0 Gauthicr-Villars

Limit points in the Lagrange spectrum of a quadratic field