Home
Hostname: page-component-78dcdb465f-bcmtx Total loading time: 0.265 Render date: 2021-04-20T02:11:37.255Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

# CONTROLLING LIPSCHITZ FUNCTIONS

Published online by Cambridge University Press:  02 August 2018

Corresponding

## Abstract

Given any positive integers $m$ and $d$ , we say a sequence of points $(x_{i})_{i\in I}$ in $\mathbb{R}^{m}$ is Lipschitz- $d$ -controlling if one can select suitable values $y_{i}\;(i\in I)$ such that for every Lipschitz function $f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$ there exists $i$ with $|f(x_{i})\,-\,y_{i}|\,<\,1$ . We conjecture that for every $m\leqslant d$ , a sequence $(x_{i})_{i\in I}\subset \mathbb{R}^{m}$ is $d$ -controlling if and only if

$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be $d$ -controlling. We also prove the conjecture for $m=1$ .

## MSC classification

Type
Research Article
Information
Mathematika , 2018 , pp. 898 - 910

## Access options

Get access to the full version of this content by using one of the access options below.

## References

Bang, Th., On covering by parallel-strips. Mat. Tidsskr. B. 1950 1950, 4953.Google Scholar
Bang, Th., A solution of the “plank problem”. Proc. Amer. Math. Soc. 2 1951, 990993.Google Scholar
Brass, P., Moser, W. and Pach, J., Research Problems in Discrete Geometry, Springer (Heidelberg, 2005).Google Scholar
Erdős, P. and Pach, J., On a problem of L. Fejes Tóth. Discrete Math. 30(2) 1980, 103109.CrossRefGoogle Scholar
Fejes Tóth, L., Remarks on the dual of Tarski’s plank problem. Mat. Lapok (N.S.) 25 1974, 1320 (in Hungarian).Google Scholar
Kupavskii, A. and Pach, J., Simultaneous approximation of polynomials. In Discrete and Computational Geometry and Graphs (Lecture Notes in Computer Science 9943 ) (eds Akiyama, J., Ito, H., Sakai, T. and Uno, Y.), Springer (Cham, 2016), 193203.CrossRefGoogle Scholar
Makai, E. Jr. and Pach, J., Controlling function classes and covering Euclidean space. Studia Sci. Math. Hungar. 18 1983, 435459.Google Scholar
McFarland, A., McFarland, J. and Smith, J. T. (eds), Alfred Tarski: Early Work in Poland–Geometry and Teaching, Birkhäuser/Springer (New York, 2014); with a bibliographic supplement, Foreword by Ivor Grattan–Guinness.CrossRefGoogle Scholar
Moese, H., Przyczynek do problemu A. Tarskiego: “O stopniu równowaonosci wielokatów”. Parametr 2 1932, 305309 (A contribution to the problem of A. Tarski, “On the degree of equivalence of polygons”).Google Scholar
Tarski, A., Uwagi o stopnii równowaznosci wielokatów. Parametr 2 1932, 310314.Google Scholar

### Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 5
Total number of PDF views: 40 *
View data table for this chart

* Views captured on Cambridge Core between 02nd August 2018 - 20th April 2021. This data will be updated every 24 hours.

# Send article to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

CONTROLLING LIPSCHITZ FUNCTIONS
Available formats
×

# Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

CONTROLLING LIPSCHITZ FUNCTIONS
Available formats
×

# Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

CONTROLLING LIPSCHITZ FUNCTIONS
Available formats
×
×