Geodesic
Diophantine approximation with partial sums of power series
Bruce C. Berndt 1   ; Sun Kim 2   ; M. Tip Phaovibul 1   ; Alexandru Zaharescu 3
1 Department of Mathematics University of Illinois 1409 West Green St. Urbana, IL 61801, U.S.A.
2 Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210, U.S.A.
3 Department of Mathematics University of Illinois 1409 West Green St. Urbana, IL 61801, U.S.A. and Institute of Mathematics of the Romanian Academy P.O. Box 1-764 Bucureşti RO-70700, Romania
Acta Arithmetica, Tome 161 (2013) no. 3, pp. 249-266 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.
DOI : 10.4064/aa161-3-4
Keywords: study question often partial sums power series functions coalesce convergents simple continued fractions functions theorems quantitatively demonstrate answer often conjecture cases there only finite number partial sums coinciding convergents many these cases offer exact numbers conjectures

Bruce C. Berndt  1   ; Sun Kim  2   ; M. Tip Phaovibul  1   ; Alexandru Zaharescu  3

1 Department of Mathematics University of Illinois 1409 West Green St. Urbana, IL 61801, U.S.A.
2 Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210, U.S.A.
3 Department of Mathematics University of Illinois 1409 West Green St. Urbana, IL 61801, U.S.A. and Institute of Mathematics of the Romanian Academy P.O. Box 1-764 Bucureşti RO-70700, Romania 
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     title = {Diophantine approximation with partial sums of power series},
     journal = {Acta Arithmetica},
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Bruce C. Berndt; Sun Kim; M. Tip Phaovibul; Alexandru Zaharescu. Diophantine approximation with partial sums of power series. Acta Arithmetica, Tome 161 (2013) no. 3, pp. 249-266. doi: 10.4064/aa161-3-4

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