| Title:
|
Some results on Poincaré sets (English) |
| Author:
|
Tang, Min-wei |
| Author:
|
Wu, Zhi-Yi |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
891-903 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is known that a set $H$ of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if $\dim _{\mathcal {H}}(X_{H})=0$, where $$ X_{H}:=\biggl \{ x=\sum ^{\infty }_{n=1} \frac {x_{n}}{2^{n}} \colon x_{n}\in \{0,1\}, x_{n} x_{n+h}=0 \ \text {for all} \ n\geq 1, \ h\in H\biggr \} $$ and $\dim _{\mathcal {H}}$ denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set $X_H$ by replacing $2$ with $b>2$. It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results. (English) |
| Keyword:
|
Poincaré set |
| Keyword:
|
homogeneous set |
| Keyword:
|
Hausdorff dimension |
| MSC:
|
11A07 |
| MSC:
|
37B20 |
| idZBL:
|
07250696 |
| idMR:
|
MR4151712 |
| DOI:
|
10.21136/CMJ.2020.0001-19 |
| . |
| Date available:
|
2020-09-07T09:42:16Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148335 |
| . |
| Reference:
|
[1] Bergelson, V., Lesigne, E.: Van der Corput sets in ${\mathbb Z}^d$.Colloq. Math. 110 (2008), 1-49. Zbl 1177.37018, MR 2353898, 10.4064/cm110-1-1 |
| Reference:
|
[2] Bishop, C. J., Peres, Y.: Fractals in Probability and Analysis.Cambridge Studies in Advanced Mathematics 162, Cambridge University Press, Cambridge (2017). Zbl 1390.28012, MR 3616046, 10.1017/9781316460238 |
| Reference:
|
[3] Bourgain, J.: Ruzsa's problem on sets of recurrence.Isr. J. Math. 59 (1987), 150-166. Zbl 0643.10045, MR 0920079, 10.1007/BF02787258 |
| Reference:
|
[4] Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications.Wiley, New York (2003). Zbl 1060.28005, MR 2118797, 10.1002/0470013850 |
| Reference:
|
[5] Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.J. Anal. Math. 31 (1977), 204-256. Zbl 0347.28016, MR 0498471, 10.1007/BF02813304 |
| Reference:
|
[6] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory.Princenton University Press, Princenton (1981). Zbl 0459.28023, MR 603625, 10.1515/9781400855162 |
| Reference:
|
[7] Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory.Graduate Texts in Mathematics 84, Springer, New York (1990). Zbl 0712.11001, MR 1070716, 10.1007/978-1-4757-2103-4 |
| Reference:
|
[8] Kamae, T., France, M. Mendès: Van der Corput's difference theorem.Isr. J. Math. 31 (1978), 335-342. Zbl 0396.10040, MR 516154, 10.1007/BF02761498 |
| Reference:
|
[9] Lê, T. H.: Problems and results on intersective sets.Combinatorial and Additive Number Theory---CANT 2011 Springer Proceedings in Mathematics & Statistics 101, Springer, New York (2014), 115-128. Zbl 1371.11028, MR 3297075, 10.1007/978-1-4939-1601-6_9 |
| Reference:
|
[10] Montgomery, H. L.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis.CBMS Regional Conference Series in Mathematics 84, American Mathematical Society, Providence (1994). Zbl 0814.11001, MR 1297543, 10.1090/cbms/084 |
| Reference:
|
[11] Ruzsa, I. Z.: Uniform distribution, positive trigonometric polynomials and difference sets.Sémin. Théor. Nombres, Univ. Bordeaux I. (1982), Article ID 18, 18 pages. Zbl 0515.10048, MR 0695335 |
| Reference:
|
[12] Sárközy, A.: On difference sets of sequences of integers I.Acta Math. Acad. Sci. Hung. 31 (1978), 125-149. Zbl 0387.10033, MR 466059, 10.1007/BF01896079 |
| Reference:
|
[13] Sárközy, A.: On difference sets of sequences of integers II.Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 21 (1978), 45-53. Zbl 0413.10051, MR 536201 |
| Reference:
|
[14] Sárközy, A.: On difference sets of sequences of integers III.Acta Math. Acad. Sci. Hung. 31 (1978), 355-386. Zbl 0387.10034, MR 487031, 10.1007/BF01901984 |
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