Construction of Minkowski Sums by Cellular Automata

TL;DR

This paper presents a cellular automaton-based method for constructing Minkowski sums of sets, including integers that are sums of three squares, providing new insights into automata and number theory.

Contribution

It introduces a novel cellular automaton construction for Minkowski sums, addressing a previously open question about sums of three squares.

Findings

Constructed Minkowski sums within cellular automata columns.

Provided a new automaton-based proof for integers as sums of three squares.

Connected cellular automata with classical number theory results.

Abstract

We give a construction in a column of a one-dimensional cellular automaton of the Minkowski sum of two sets which can themselves occur in columns of cellular automata. It enables us to obtain another construction of the set of integers that are sums of three squares, answering a question by the same author.