## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 58, Number 2 (2018), 187-205.

### MSTD sets and Freiman isomorphisms

#### Abstract

An MSTD set is a finite set with more pairwise sums than differences. $(\Upsilon,\Phi)$-ismorphisms are generalizations of Freiman isomorphisms to arbitrary linear forms. These generalized isomorphisms are used to prove that every finite set of real numbers is Freiman isomorphic to a finite set of integers. This implies that there exists no MSTD set $A$ of real numbers with $|A| \leq 7$, and, up to Freiman isomorphism and affine isomorphism, there exists exactly one MSTD set $A$ of real numbers with $|A| = 8$.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 58, Number 2 (2018), 187-205.

**Dates**

First available in Project Euclid: 2 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1512183760

**Digital Object Identifier**

doi:10.7169/facm/1685

**Mathematical Reviews number (MathSciNet)**

MR3816073

**Zentralblatt MATH identifier**

06924926

**Subjects**

Primary: 11B13: Additive bases, including sumsets [See also 05B10]

Secondary: 11B75: Other combinatorial number theory 05B20: Matrices (incidence, Hadamard, etc.) 05A19: Combinatorial identities, bijective combinatorics 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 11D04: Linear equations

**Keywords**

MSTD set Freiman isomorphism $(\Upsilon,\Phi)$-ismorphism sumset difference set linear forms Dirichlet's theorem product set quotient set MPTQ set

#### Citation

Nathanson, Melvyn B. MSTD sets and Freiman isomorphisms. Funct. Approx. Comment. Math. 58 (2018), no. 2, 187--205. doi:10.7169/facm/1685. https://projecteuclid.org/euclid.facm/1512183760