Functiones et Approximatio Commentarii Mathematici

MSTD sets and Freiman isomorphisms

Melvyn B. Nathanson

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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


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