Density and finiteness results on sums of fractions

Göral, Haydar; Sertbaş, Doğa Can

doi:10.1090/proc/14270

Density and finiteness results on sums of fractions
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Proc. Amer. Math. Soc. 147 (2019), 567-581 Request permission

Abstract:

We show that the height density of a finite sum of fractions is zero. In fact, we give quantitative estimates in terms of the height function. Then, we focus on the unit fraction solutions in the ring of integers of a given number field. In particular, we prove that finitely many representations of 1 as a sum of unit fractions determines the field of rational numbers among all real number fields. Finally, using non-standard methods, we prove some density and finiteness results on a finite sum of fractions.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
T. D. Browning and C. Elsholtz, The number of representations of rationals as a sum of unit fractions, Illinois J. Math. 55 (2011), no. 2, 685–696 (2012). MR 3020702
D. R. Curtiss, On Kellogg’s Diophantine Problem, Amer. Math. Monthly 29 (1922), no. 10, 380–387. MR 1520110, DOI 10.2307/2299023
Lou van den Dries and Ayhan Günaydın, The fields of real and complex numbers with a small multiplicative group, Proc. London Math. Soc. (3) 93 (2006), no. 1, 43–81. MR 2235481, DOI 10.1017/S0024611506015747
Christian Elsholtz, Sums of $k$ unit fractions, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3209–3227. MR 1828604, DOI 10.1090/S0002-9947-01-02782-9
P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 28, Université de Genève, L’Enseignement Mathématique, Geneva, 1980. MR 592420
J.-H. Evertse, H. P. Schlickewei, and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2) 155 (2002), no. 3, 807–836. MR 1923966, DOI 10.2307/3062133
Robert Goldblatt, Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998. An introduction to nonstandard analysis. MR 1643950, DOI 10.1007/978-1-4612-0615-6
H. Göral, Model theory of fields and heights, Ph.D. thesis, Lyon, 2015.
Haydar Göral, Height bounds, Nullstellensatz and primality, Comm. Algebra 46 (2018), no. 10, 4463–4472. MR 3847127, DOI 10.1080/00927872.2018.1448835
H. Göral, Tame expansions of $\omega$-stable theories and definable groups, to appear, Notre Dame Journal of Formal Logic.
Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR 1299330, DOI 10.1007/978-1-4899-3585-4
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
Gerd Hofmeister and Peter Stoll, Note on Egyptian fractions, J. Reine Angew. Math. 362 (1985), 141–145. MR 809971
Jing-Jing Huang and Robert C. Vaughan, On the exceptional set for binary Egyptian fractions, Bull. Lond. Math. Soc. 45 (2013), no. 4, 861–874. MR 3081553, DOI 10.1112/blms/bdt020
S. V. Konyagin, Double exponential lower bound for the number of representations of unity by Egyptian fractions, Math. Notes 95 (2014), no. 1-2, 277–281. Translation of Mat. Zametki 95 (2014), no. 2, 312–316. MR 3267215, DOI 10.1134/S0001434614010295
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
M. B. Nathanson, Weighted real Egyptian numbers, preprint, https://arxiv.org/pdf/1708.09478
Pierre Samuel, Algebraic theory of numbers, Houghton Mifflin Co., Boston, Mass., 1970. Translated from the French by Allan J. Silberger. MR 0265266
André Schinzel, Sur quelques propriétés des nombres $3/n$ et $4/n$, où $n$ est un nombre impair, Mathesis 65 (1956), 219–222 (French). MR 80683
W. Sierpiński, Sur les décompositions de nombres rationnels en fractions primaires, Mathesis 65 (1956), 16–32 (French). MR 78385
K. Soundararajan, Approximating 1 from below using Egyptian fractions, 2005, preprint, https://arxiv.org/pdf/math/0502247.pdf.
J. J. Sylvester, On a Point in the Theory of Vulgar Fractions, Amer. J. Math. 3 (1880), no. 4, 332–335. MR 1505274, DOI 10.2307/2369261
R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), 193–198. MR 289409, DOI 10.1112/S0025579300002886
C. Viola, On the diophantine equations $\Pi ^{k}_{0}x_{i}-\sum ^{k}_{0}\,x_{i}=n$ and $\sum ^{k}_{0}\,1/x_{i}=a/n$, Acta Arith. 22 (1972/73), 339–352. MR 321869, DOI 10.4064/aa-22-3-339-352
Hisashi Yokota, On the number of integers representable as sums of unit fractions. III, J. Number Theory 96 (2002), no. 2, 351–372. MR 1932461