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Keywords:
Engel expansion; finite pattern; upper Banach density; arithmetic progression; geometric progression
Engel expansion; finite pattern; upper Banach density; arithmetic progression; geometric progression
Summary:
Let $\mathcal {F}$ be a countable collection of functions $f$ defined on integers, with integer values, such that for every $f\in \mathcal {F}$, $f(n)\to \infty $ as $n\to \infty $. This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain every finite pattern of $\mathcal {F}$, with applications demonstrated through representative examples.
References:
[1] Billingsley, P.: Hausdorff dimension in probability theory. II. Ill. J. Math. 5 (1961), 291-298. DOI 10.1215/ijm/1255629826 | MR 0120339 | Zbl 0098.10602
[2] Dyatlov, S., Zahl, J.: Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26 (2016), 1011-1094. DOI 10.1007/s00039-016-0378-3 | MR 3558305 | Zbl 1384.58019
[3] Falconer, K., Yavicoli, A.: Intersections of thick compact sets in $\Bbb{R}^d$. Math. Z. 301 (2022), 2291-2315. DOI 10.1007/s00209-022-02992-y | MR 4437323 | Zbl 1514.11017
[4] Fan, A., Wang, B., Wu, J.: Arithmetic and metric properties of Oppenheim continued fraction expansions. J. Number Theory 127 (2007), 64-82. DOI 10.1016/j.jnt.2006.12.016 | MR 2351664 | Zbl 1210.11086
[5] Fraser, J. M., Saito, K., Yu, H.: Dimensions of sets which uniformly avoid arithmetic progressions. Int. Math. Res. Not. 2019 (2019), 4419-4430. DOI 10.1093/imrn/rnx261 | MR 3984074 | Zbl 1460.28008
[6] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, New Jersey (1981). MR 0603625 | Zbl 0459.28023
[7] Galambos, J.: Representations of Real Numbers by Infinite Series. Lecture Notes in Mathematics 502. Springer, New York (1976). DOI 10.1007/BFb0081642 | MR 0568141 | Zbl 0322.10002
[8] Green, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Ann. Math. (2) 167 (2008), 481-547. DOI 10.4007/annals.2008.167.481 | MR 2415379 | Zbl 1191.11025
[9] Łaba, I., Pramanik, M.: Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19 (2009), 429-456. DOI 10.1007/s00039-009-0003-9 | MR 2545245 | Zbl 1184.28010
[10] Lai, C.-K.: Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions. Ann. Acad. Sci. Fenn., Math. 42 (2017), 1009-1017. DOI 10.5186/aasfm.2017.4263 | MR 3701662 | Zbl 1403.28006
[11] Molter, U., Yavicoli, A.: Small sets containing any pattern. Math. Proc. Camb. Philos. Soc. 168 (2020), 57-73. DOI 10.1017/S0305004118000567 | MR 4043821 | Zbl 1429.28008
[12] Potgieter, P.: Arithmetic progressions in Salem-type subsets of the integers. J. Fourier Anal. Appl. 17 (2011), 1138-1151. DOI 10.1007/s00041-011-9179-0 | MR 2854833 | Zbl 1270.42007
[13] Song, K., Li, Z., Shang, L.: On the growth rate of partial quotients in Engel continued fractions. Publ. Math. Debr. 106 (2025), 103-123. DOI 10.5486/PMD.2025.9822 | MR 4859312 | Zbl 07985759
[14] Szemerédi, E.: On sets of integers containing no $k$ elements in arithmetic progression. Acta Arith. 27 (1975), 199-245. DOI 10.4064/aa-27-1-199-245 | MR 0369312 | Zbl 0303.10056
[15] Tian, Z., Fang, L.: On arithmetic progressions in the digits of Engel expansions. Fractals 33 (2025), Article ID 2450145. DOI 10.1142/S0218348X24501457
[16] Tian, Z., Wu, M., Lou, M.-L.: Finite pattern problems related to Lüroth expansion. Fractals 28 (2020), Article ID 2050048, 6 pages. DOI 10.1142/S0218348X20500486 | Zbl 1434.11156
[17] Tong, X., Wang, B.: How many points contain arithmetic progressions in their continued fraction expansion?. Acta Arith. 139 (2009), 369-376. DOI 10.4064/aa139-4-4 | MR 2545935 | Zbl 1205.11092
[18] Xi, L.-F.: Quasi-Lipschitz equivalence of fractals. Isr. J. Math. 160 (2007), 1-21. DOI 10.1007/s11856-007-0053-3 | MR 2342488 | Zbl 1145.28007
[19] Xi, L., Jiang, K., Pei, Q.: Arithmetic progressions in self-similar sets. Front. Math. China 14 (2019), 957-966. DOI 10.1007/s11464-019-0788-2 | MR 4025036 | Zbl 1487.28015
[20] Yavicoli, A.: Patterns in thick compact sets. Isr. J. Math. 244 (2021), 95-126. DOI 10.1007/s11856-021-2173-6 | MR 4344022 | Zbl 1483.28012
[21] Zhang, Z., Cao, C.: On points contain arithmetic progressions in their Lüroth expansion. Acta Math. Sci., Ser. B, Engl. Ed. 36 (2016), 257-264. DOI 10.1016/S0252-9602(15)30093-X | MR 3432763 | Zbl 1363.11080
[22] Zhang, Z.-L., Cao, C.-Y.: On points with positive density of the digit sequence in infinite iterated function systems. J. Aust. Math. Soc. 102 (2017), 435-443. DOI 10.1017/S1446788716000288 | MR 3650967 | Zbl 1428.11141
[23] Zhao, X., Shen, L.: Localized growth speed of the digits in Engel expansions. J. Math. Anal. Appl. 530 (2024), Article ID 127657, 9 pages. DOI 10.1016/j.jmaa.2023.127657 | MR 4632747 | Zbl 1535.11113
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