Korean J. Math. Vol. 25 No. 2 (2017) pp.137-145
DOI: https://doi.org/10.11568/kjm.2017.25.2.137

Continued fractions and the density of graphs of some functions

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Published: Jun 30, 2017
Keywords:
continued fractions, Dedekind sums
Mathematics Subject Classification:
11B05, 11J70, 11J71

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Hi-joon Chae
Department of Mathematics Education Hongik University Seoul 04066, Republic of Korea
Byungheup Jun
Department of Mathematical Sciences UNIST Ulsan 44919, Republic of Korea
Jungyun Lee
Department of Mathematics Ewha Womans University Seoul 03760, Republic of Korea

Abstract

We consider some simple periodic functions on the field of rational numbers with values in $\mathbb{Q}/\mathbb{Z}$ which are defined in terms of lowest-term-expression of rational numbers. We prove the density of graphs of these functions by constructing explicitly points on the graphs close to a given point using continued fractions.



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Supporting Agencies

This work was supported by 2014 Hongik University Research Fund NRF-2015R1D1A1A09059083 NRF-2009-0093827.

References

[1] H. Chae, B. Jun and J. Lee, Equidistribution of higher dimensional generalized Dedekind sums and exponential sums, Submitted. Google Scholar

[2] J. Lee, B. Jun and H. Chae, Higher Hickerson formula, J. Number Theory 170 (2017), 191–210. Google Scholar

[3] H. Davenport and P. Erd ̈os, The distribution of quadratic and higher residues, Publ. Math. Debrecen 2 (1952), 252–265. Google Scholar

[4] D. Hickerson, Continued fractions and density results for Dedekind sums, J. Reine Angew. Math. 290 (1977), 113–116. Google Scholar

[5] H. Rademacher and E. Grosswald, Dedekind sums, Carus Math. Monogr. 16, Math. Assoc. Amer., 1972. Google Scholar

[6] T. Tao, Higher order Fourier analysis, Graduate Studies in Mathematics 142, Amer. Math. Soc., 2012. Google Scholar