Some results on Poincaré sets

Min-wei Tang; Zhi-Yi Wu

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 3, page 891-903
  • ISSN: 0011-4642

Abstract

top
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 ℋ ( X H ) = 0 , where X H : = x = ∑ n = 1 ∞ x n 2 n : x n ∈ { 0 , 1 } , x n x n + h = 0 for all n ≥ 1 , h ∈ H and dim 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.

How to cite

top

Tang, Min-wei, and Wu, Zhi-Yi. "Some results on Poincaré sets." Czechoslovak Mathematical Journal 70.3 (2020): 891-903. <http://eudml.org/doc/297222>.

@article{Tang2020,
abstract = {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 \[ X\_\{H\}:=\biggl \lbrace x=\sum ^\{\infty \}\_\{n=1\} \frac\{x\_\{n\}\}\{2^\{n\}\} \colon x\_\{n\}\in \lbrace 0,1\rbrace , x\_\{n\} x\_\{n+h\}=0 \ \text\{for all\} \ n\ge 1, \ h\in H\biggr \rbrace \] 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.},
author = {Tang, Min-wei, Wu, Zhi-Yi},
journal = {Czechoslovak Mathematical Journal},
keywords = {Poincaré set; homogeneous set; Hausdorff dimension},
language = {eng},
number = {3},
pages = {891-903},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some results on Poincaré sets},
url = {http://eudml.org/doc/297222},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Tang, Min-wei
AU - Wu, Zhi-Yi
TI - Some results on Poincaré sets
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 891
EP - 903
AB - 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 \[ X_{H}:=\biggl \lbrace x=\sum ^{\infty }_{n=1} \frac{x_{n}}{2^{n}} \colon x_{n}\in \lbrace 0,1\rbrace , x_{n} x_{n+h}=0 \ \text{for all} \ n\ge 1, \ h\in H\biggr \rbrace \] 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.
LA - eng
KW - Poincaré set; homogeneous set; Hausdorff dimension
UR - http://eudml.org/doc/297222
ER -

References

top
  1. Bergelson, V., Lesigne, E., 10.4064/cm110-1-1, Colloq. Math. 110 (2008), 1-49. (2008) Zbl1177.37018MR2353898DOI10.4064/cm110-1-1
  2. Bishop, C. J., Peres, Y., 10.1017/9781316460238, Cambridge Studies in Advanced Mathematics 162, Cambridge University Press, Cambridge (2017). (2017) Zbl1390.28012MR3616046DOI10.1017/9781316460238
  3. Bourgain, J., 10.1007/BF02787258, Isr. J. Math. 59 (1987), 150-166. (1987) Zbl0643.10045MR0920079DOI10.1007/BF02787258
  4. Falconer, K., 10.1002/0470013850, Wiley, New York (2003). (2003) Zbl1060.28005MR2118797DOI10.1002/0470013850
  5. Furstenberg, H., 10.1007/BF02813304, J. Anal. Math. 31 (1977), 204-256. (1977) Zbl0347.28016MR0498471DOI10.1007/BF02813304
  6. Furstenberg, H., 10.1515/9781400855162, Princenton University Press, Princenton (1981). (1981) Zbl0459.28023MR603625DOI10.1515/9781400855162
  7. Ireland, K., Rosen, M., 10.1007/978-1-4757-2103-4, Graduate Texts in Mathematics 84, Springer, New York (1990). (1990) Zbl0712.11001MR1070716DOI10.1007/978-1-4757-2103-4
  8. Kamae, T., France, M. Mendès, 10.1007/BF02761498, Isr. J. Math. 31 (1978), 335-342. (1978) Zbl0396.10040MR516154DOI10.1007/BF02761498
  9. Lê, T. H., 10.1007/978-1-4939-1601-6_9, Combinatorial and Additive Number Theory---CANT 2011 Springer Proceedings in Mathematics & Statistics 101, Springer, New York (2014), 115-128. (2014) Zbl1371.11028MR3297075DOI10.1007/978-1-4939-1601-6_9
  10. Montgomery, H. L., 10.1090/cbms/084, CBMS Regional Conference Series in Mathematics 84, American Mathematical Society, Providence (1994). (1994) Zbl0814.11001MR1297543DOI10.1090/cbms/084
  11. Ruzsa, I. Z., Uniform distribution, positive trigonometric polynomials and difference sets, Sémin. Théor. Nombres, Univ. Bordeaux I. (1982), Article ID 18, 18 pages. (1982) Zbl0515.10048MR0695335
  12. Sárközy, A., 10.1007/BF01896079, Acta Math. Acad. Sci. Hung. 31 (1978), 125-149. (1978) Zbl0387.10033MR466059DOI10.1007/BF01896079
  13. Sárközy, A., On difference sets of sequences of integers II, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 21 (1978), 45-53. (1978) Zbl0413.10051MR536201
  14. Sárközy, A., 10.1007/BF01901984, Acta Math. Acad. Sci. Hung. 31 (1978), 355-386. (1978) Zbl0387.10034MR487031DOI10.1007/BF01901984

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.