Third Italian Number Theory Meeting

# Carries and the arithmetic progression structure of sets

speaker: Francesco Monopoli (Università degli Studi di Milano)

abstract: If we want to represent integers in base $$m$$, we need a set $$A$$ of digits, which needs to be a complete set of residues modulo $$m$$. When adding two integers with last digits $$a_1, a_2 \in A$$, we find the unique $$a \in A$$ such that $$a_1 + a_2 \equiv a$$ mod $$m$$, and call $$(a_1 + a_2 -a)/m$$ the carry. Carries occur also when addition is done modulo $$m^2$$, with $$A$$ chosen as a set of coset representatives for the cyclic group $$\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$$. It is a natural to look for sets $$A$$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $$m=p$$, $$p$$ prime, the only set $$A$$ which induces two distinct carries, i. e. with $$A+A \subseteq \{ x, y \}+A$$ for some $$x, y \in \mathbb{Z}/p^2\mathbb{Z}$$, is the arithmetic progression $$[0, p-1]$$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $$m^2$$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference. (Joint work with Imre Z. Ruzsa)

timetable:
Tue 22 Sep, 15:30 - 15:55, Aula Russo
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