Abstract
Erdős proved that any real number can be written as a sum, and a product, of two Liouville numbers. Motivated by these results, we study sumsets of classes of real numbers with prescribed (or bounded) irrationality exponents. We show that such sumsets turn out to be large in general, indeed almost every real number with respect to Lebesgue measure can be written as the sum of two numbers with sufficiently large prescribed irrationality exponents. In fact the Hausdorff dimension of the complement is small, and the result remains true if we impose considerably refined conditions on the orders of rational approximation (“exact approximation” with respect to an approximation function). As an application, we show that in many cases the Hausdorff dimension of Cartesian products of sets with prescribed irrationality exponent exceeds the expected dimension, that is the sum of the single Hausdorff dimensions. We also address their packing dimensions. Similar results hold when restricting to classical missing digit Cantor sets, relative to its natural Cantor measure. In particular, we prove that the subset of numbers with prescribed large irrationality exponent has full packing dimension, i.e. the same packing dimension as the entire Cantor set. This complements some results on the Hausdorff dimension of these sets, which is an extensively studied topic in Diophantine approximation. Our proofs are based on ideas of Erdős, but vastly extend them.
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References
Akhunzhanov, R.: Vectors of a given Diophantine type II. Mat. Sb. 204(4), 1–22 (2013)
Akhunzhanov, R., Moshchevitin, N.: Vectors of a given Diophantine type. Mat. Zam. 80(3), 328–338 (2006) (Russian). English translation Math. Notes 80(3–4), 318–328 (2006)
Alniaçik, K.: On Mahler’s \(U\)-numbers. Am. J. Math. 105(6), 1347–1356 (1983)
Amou, M., Bugeaud, Y.: Exponents of Diophantine approximation and expansions in integer bases. J. Lond. Math. Soc. 81(2), 297–316 (2010)
Besicovitch, A.S., Moran, P.A.P.: The measure of product and cylinder sets. J. Lond. Math. Soc. 20, 110–120 (1945)
Bishop, C.J., Peres, Y.: Packing dimension and Cartesian products. Trans. Am. Math. Soc. 348(11), 4433–4445 (1996)
Borosh, I., Fraenkel, I.A.S.: A generalization of Jarník’s theorem on Diophantine approximations. Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34, 193–201 (1972)
Bugeaud, Y.: Sets of exact approximation order by rational numbers. Math. Ann. 327(1), 171–190 (2003)
Bugeaud, Y.: Diophantine approximation and Cantor sets. Math. Ann. 341(3), 677–684 (2008)
Bugeaud, Y., Dodson, M.M., Kristensen, S.: Zero-infinity laws in Diophantine approximation. Q. J. Math. 56(3), 311–320 (2005)
Bugeaud, Y., Durand, A.: Metric Diophantine approximation on the middle-third Cantor set. J. Eur. Math. Soc. (JEMS) 18(6), 1233–1272 (2016)
Burger, E.B.: On Liouville decompositions in local fields. Proc. Am. Math. Soc. 124(11), 3305–3310 (1996)
Burger, E.B.: Diophantine inequalities and irrationality measures for certain transcendental numbers. Indian J. Pure Appl. Math. 32(10), 1591–1599 (2001)
Chalebgwa, T.P., Morris, S.A.: Erdős–Liouville sets. Bull. Aust. Math. Soc. 107(2), 284–289 (2023)
Chalebgwa, T.P., Morris, S.A.: Measures of sets of transcendental numbers and the Erdős property. Submitted
Das, T., Fishman, L., Simmons, D., Urbański, M.: A variational principle in the parametric geometry of numbers. arXiv:1901.06602
Eggleston, H.G.: A correction to a paper on the dimension of cartesian product sets. Proc. Camb. Philos. Soc. 49, 437–440 (1953)
Erdős, P.: Representations of real numbers as sums and products of Liouville numbers. Mich. Math. J. 9, 59–60 (1962)
Falconer, K.: Fractal geometry. In: Mathematical Foundations and Applications. Wiley, Chichester (1990)
Hussain, M., Schleischitz, J., Simmons, D.: The generalised Baker–Schmidt problem on hypersurfaces. Int. Math. Res. Not. IMRN 12, 8845–8867 (2021)
Jarník, V.: Über die simultanen diophantischen Approximationen. Math. Z. 33, 505–543 (1931)
Koivusalo, H., Levesley, J., Ward, B., Zhang, X.: The dimension of the set of \(\Psi \)-badly approximable points in all ambient dimensions; on a question of Beresnevich and Velani. In preparation
Larman, D.G.: On Hausdorff measure in finite-dimensional compact metric spaces. Proc. Lond. Math. Soc. 17(3), 193–206 (1967)
Levesley, J., Salp, C., Velani, S.L.: On a problem of K. Mahler: Diophantine approximation and Cantor sets. Math. Ann. 338(1), 97–118 (2007)
Marnat, A.: Hausdorff and packing dimension of Diophantine sets. arXiv:1904.08416
Marstrand, J.M.: The dimension of Cartesian product sets. Proc. Camb. Philos. Soc. 50, 198–202 (1954)
Olsen, L., Renfro, D.L.: On the exact Hausdorff dimension of the set of Liouville numbers II. Manuscr. Math. 119(2), 217–224 (2006)
Oxtoby, J.: Measure and category. In: A Survey of the Analogies Between Topological and Measure Spaces. Graduate Texts in Mathematics, vol. 2, 2edn. Springer, New York (1980)
Perron, O.: Die Lehre von den Kettenbrüchen. B.G. Teubner (1913)
Petruska, G.: On strong Liouville numbers. Indag. Math. (N.S.) 3, 211–218 (1992)
Rieger, G.J.: Über die Lösbarkeit von Gleichungssystemen durch Liouville-Zahlen. (German) Arch. Math. (Basel) 26, 40–43 (1975)
Schleischitz, J.: Generalizations of a result of Jarník on simultaneous approximation. Mosc. J. Comb. Number Theory 6(2–3), 253–287 (2016)
Schleischitz, J.: Rational approximation to surfaces defined by polynomials in one variable. Acta Math. Hungar. 155(2), 362–375 (2018)
Senthil Kumar, K., Thangadurai, R., Waldschmidt, M.: Liouville numbers and Schanuel’s Conjecture. Arch. Math. (Basel) 102(1), 59–70 (2014)
Schwarz, W.: Liouville–Zahlen und der Satz von Baire. Math. Phys. Semesterber. 24(1), 84–87 (1977) (German)
Tricot, C., Jr.: Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91(1), 57–74 (1982)
Wegmann, H.: Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine Angew. Math. 234, 163–171 (1969)
Weiss, B.: Almost no points on a Cantor set are very well approximable. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2008), 949–952 (2001)
Xiao, Y.: Packing dimension, Hausdorff dimension and Cartesian product sets. Math. Proc. Camb. Philos. Soc. 120(3), 535–546 (1996)
Yu, H.: Rational points near self-similar sets. arXiv:2101.05910
Acknowledgements
The author thanks the referee for the careful reading. I am further thankful to Sidney A. Morris for bringing to my attention his joint paper with Chalebgwa on Erdos-Liouville sets.
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Schleischitz, J. Metric Results on Sumsets and Cartesian Products of Classes of Diophantine Sets. Results Math 78, 215 (2023). https://doi.org/10.1007/s00025-023-02000-7
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DOI: https://doi.org/10.1007/s00025-023-02000-7