## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 56, Number 4 (2012), 1297-1312.

### Circle discrepancy for checkerboard measures

Mihail N. Kolountzakis and Ioannis Parissis

#### Abstract

Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius $t\geq1$ there exists a full circle of radius either $t$ or $2t$ with discrepancy greater than $c\sqrt{t}$ for some numerical constant $c>0$. We also show that for every $t\geq1$ there exists a circular arc of radius exactly $t$ with discrepancy greater than $c\sqrt{t}$. Finally, we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.

#### Article information

**Source**

Illinois J. Math., Volume 56, Number 4 (2012), 1297-1312.

**Dates**

First available in Project Euclid: 6 May 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1399395833

**Digital Object Identifier**

doi:10.1215/ijm/1399395833

**Mathematical Reviews number (MathSciNet)**

MR3231484

**Zentralblatt MATH identifier**

1370.11094

**Subjects**

Primary: 11K31: Special sequences 11K38: Irregularities of distribution, discrepancy [See also 11Nxx]

#### Citation

Kolountzakis, Mihail N.; Parissis, Ioannis. Circle discrepancy for checkerboard measures. Illinois J. Math. 56 (2012), no. 4, 1297--1312. doi:10.1215/ijm/1399395833. https://projecteuclid.org/euclid.ijm/1399395833