Attainable measures for certain types of $p$-adic Duffin-Schaeffer sets
DOI:
https://doi.org/10.7146/math.scand.a-139832Abstract
This paper settles recent conjectures concerning the $p$-adic Haar measure applied to a family of sets defined in terms of Diophantine approximation. This is done by determining the spectrum of measure values for each family and seeing that this contradicts the corresponding conjectures.
References
Badziahin, D., and Bugeaud, Y., Multiplicative $p$-adic approximation, Michigan Math. J. 71 (2022), no. 1, 121–143. https://doi.org/10.1307/mmj/20195785
Cassels, J. W. S., Some metrical theorems in Diophantine approximation. I, Proc. Cambridge Philos. Soc. 46 (1950), no. 2, 209–218. https://doi.org/10.1017/S0305004100025676
Duffin, R. J., and Schaeffer, A. C., Khintchine's problem in metric Diophantine approximation, Duke Math. J. 8 (1941), no. 2, 243–255. https://doi.org/10.1215/S0012-7094-41-00818-9
Gallagher, P., Approximation by reduced fractions, J. Math. Soc. Japan 13 (1961), no. 4, 342–345. https://doi.org/10.2969/jmsj/01340342
Haynes, A. K., The metric theory of $p$-adic approximation, Int. Math. Res. Not. IMRN 2010, no. 1, 18–52. https://doi.org/10.1093/imrn/rnp114
Jarník, V., Sur les approximations diophantiques des nombres $p$-adiques, Rev. Ci. (Lima) 47 (1945), 489–505.
Khintchine, A., Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), no. 1–2, 115–125. https://doi.org/10.1007/BF01448437
Koukoulopoulos, D., and Maynard, J., On the Duffin-Schaeffer conjecture, Ann. of Math. (2) 192 (2020), no. 1, 251–307. https://doi.org/10.4007/annals.2020.192.1.5
Kristensen, S., and Laursen, M. L., The $p$-adic Duffin-Schaeffer conjecture, Funct. Approx. Comment. Math. 68 (2023), no. 1, 113–126. https://doi.org/10.7169/facm/2042
Lutz, E., Sur les approximations diophantiennes linéaires $p$-adiques, Actualités Scientifiques et Industrielles; No. 1224. Hermann & Cie, Paris, 1955.
Montgomery, H. L., and Vaughan, R. C., Multiplicative number theory I. Classic theory, Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
