<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>X</italic> and <italic>Y</italic> be nonempty finite subsets of 𝕑 and <italic>X</italic> +<italic>Y</italic> its sumset. The structures of <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ):= |<italic>X</italic> +<italic>Y</italic> |−|<italic>X</italic>|−|<italic>Y</italic> | is small have been widely studied; in particular the Generalized Freiman’s 3<italic>k</italic> − 4 Theorem describes <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ) ≤ min{|<italic>X</italic>|, |<italic>Y</italic> |} − 4. However, not too much is known about <italic>X</italic> and <italic>Y</italic> when <italic>r</italic>(<italic>X, Y</italic> ) <italic>></italic> min{|<italic>X</italic>|, |<italic>Y</italic> |} − 4. In this paper we study the structure of <italic>X</italic> and <italic>Y</italic> for arbitrary <italic>r</italic>(<italic>X, Y</italic> ).</p></abstract>

Let X and Y be nonempty finite subsets of 𝕑 and X +Y its sumset. The structures of X and Y when r(X, Y ):= |X +Y |−|X|−|Y | is small have been widely studied; in particular the Generalized Freiman’s 3k − 4 Theorem describes X and Y when r(X, Y ) ≤ min{|X|, |Y |} − 4. However, not too much is known about X and Y when r(X, Y ) > min{|X|, |Y |} − 4. In this paper we study the structure of X and Y for arbitrary r(X, Y ).

On Freiman’s 3<italic>k</italic> − 4 Theorem

Institute of Physics of the Autonomous University of San Luis Potosí

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":{"italic":"k","content":["On Freiman\u2019s 3","− 4 Theorem"]}}

Let X and Y be nonempty finite subsets of 𝕑 and X +Y its sumset. The structures of X and Y when r(X, Y ):= |X +Y |−|X|−|Y | is small have been widely studied;...

On Freiman’s 3k − 4 Theorem

Published Online: Mar 27, 2020

Page range: 43 - 68

Received: Apr 19, 2018

Accepted: May 27, 2019

DOI: https://doi.org/10.2478/udt-2019-0013

Keywords
Freiman’s 3 − 4 Theorem, sumset, Lev-Smeliansky Theorem

© 2019 Mario Huicochea, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On Freiman’s 3k − 4 Theorem

Published Online: Mar 27, 2020

Page range: 43 - 68

Received: Apr 19, 2018

Accepted: May 27, 2019

DOI: https://doi.org/10.2478/udt-2019-0013

KeywordsFreiman’s 3 − 4 Theorem, sumset, Lev-Smeliansky Theorem

© 2019 Mario Huicochea, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Freiman’s 3 − 4 Theorem, sumset, Lev-Smeliansky Theorem