<abstract xmlns="http://www.w3.org/1999/xhtml">The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved <italic>inter alia</italic> that empirical distribution of the points (<italic>k,m</italic>)<italic>p</italic>−<italic>m</italic> with 0 ≤ <italic>k</italic> ≤ <italic>n &lt; pm</italic> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0017_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>≡</mo><mi>a</mi><mo> </mo><msup><mrow><mo>(</mo><mrow><mi>mod</mi><mo>⁡</mo><mtext> </mtext><mi>p</mi></mrow><mo>)</mo></mrow><mi>s</mi></msup></math><tex-math>$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $</tex-math></alternatives></inline-formula> (for (<italic>a, p</italic>) = 1) for <italic>m</italic>→∞ tends to the Hausdorff measure on the “<italic>p</italic>-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.</abstract>

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)p−m with 0 ≤ k ≤ n < pm and (nk)≡a (mod⁡ p)s$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

Institut für Analysis und Zahlentheorie, Technische Universität Graz, Kopernikusgasse 24, 8010 Graz,

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers"}

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the...

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

Publié en ligne: 13 janv. 2017

Pages: 151 - 161

Reçu: 10 mai 2016

Accepté: 14 juin 2016

DOI: https://doi.org/10.1515/udt-2016-0017

Mots clés
Binomial coefficients, equidistribution

© 2016 Guy Barat et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

Publié en ligne: 13 janv. 2017

Pages: 151 - 161

Reçu: 10 mai 2016

Accepté: 14 juin 2016

DOI: https://doi.org/10.1515/udt-2016-0017

Mots clésBinomial coefficients, equidistribution

© 2016 Guy Barat et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
Binomial coefficients, equidistribution