Abstract
We review the basic theory of more sums than differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and “explicit” constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are k-generational sum-dominant (meaning A, A + A, \(\ldots\), \(kA = A + \cdots + A\) are each sum-dominant).
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Notes
- 1.
Note that \(A + A + A - A = -(A - A - A - A)\); thus we might as well assume any linear combination has at least as many sums of A as differences of A.
- 2.
Requiring 0, 2n − 1 ∈ A is quite mild; we do this so that we know the first and last elements of A.
- 3.
As before, requiring 0, 2n − 1 ∈ A is quite mild and is done so that we know the first and last elements of A.
References
J.M. Deshouillers, G. Effinger, H. Te Riele, D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electron. Res. Announc. Am. Math. Soc. 3, 99–104 (1997)
G.A. Freiman, V.P. Pigarev, The relation between the invariants R and T. in Number Theoretic Studies in the Markov Spectrum and in the Structural Theory of Set Addition (Russian) (Kalinin Gos. University, Moscow, 1973), pp. 172–174
P.V. Hegarty, Some explicit constructions of sets with more sums than differences. Acta Arithmetica 130(1), 61–77 (2007)
P.V. Hegarty, S.J. Miller, When almost all sets are difference dominated. Random Struct. Algorithms 35(1), 118–136 (2009)
P.V. Hegarty, S.J. Miller, Appendix 2 of explicit constructions of infinite families of MSTD sets (by S. J. Miller and D. Scheinerman), in Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson, ed. by D. Chudnovsky, G. Chudnovsky (Springer, New York, 2010)
G. Iyer, O. Lazarev, S.J. Miller, L. Zhang, Generalized more sums than differences sets. J. Number Theory 132(5), 1054–1073 (2012)
J.H. Kim, V.H. Vu, Concentration of multivariate polynomials and its applications. Combinatorica 20, 417–434 (2000)
J. Marica, On a conjecture of conway. Can. Math. Bull. 12, 233–234 (1969)
G. Martin, K. O’Bryant, Many sets have more sums than differences, in Additive Combinatorics. CRM Proceedings and Lecture Notes, vol. 43 (American Mathematical Society, Providence, 2007), pp. 287–305
S.J. Miller, B. Orosz, D. Scheinerman, Explicit constructions of infinite families of MSTD sets. J. Number Theory 130, 1221–1233 (2010)
S.J. Miller, S. Pegado, S.L. Robinson, Explicit constructions of large families of generalized more sums than differences sets. Integers 12, #A30 (2012)
M.B. Nathanson, Additive Number Theory: The Classical Bases. Graduate Texts in Mathematics (Springer, New York, 1996)
M.B. Nathanson, Problems in additive number theory I, in Additive Combinatorics. CRM Proceedings and Lecture Notes, vol. 43 (American Mathematical Society, Providence, 2007), pp. 263–270
M.B. Nathanson, Sets with more sums than differences. Integers Electron. J. Combin. Number Theory 7, Paper A5 (24 pp.) (2007)
I.Z. Ruzsa, On the cardinality of A + A and A − A, in Combinatorics year (Keszthely, 1976). Coll. Math. Soc. J. Bolyai, vol. 18 (North-Holland-Bolyai T \(\grave{\mathrm{a}}\) rsulat, Budaset, 1978), pp. 933–938
I.Z. Ruzsa, Sets of sums and differences, in S \(\acute{\mathrm{e}}\) minaire de Th \(\acute{\mathrm{e}}\) orie des Nombres de Paris 1982–1983 (Birkh \(\ddot{\mathrm{a}}\) user, Boston, 1984), pp. 267–273
I.Z. Ruzsa, On the number of sums and differences. Acta Math. Sci. Hungar. 59, 439–447 (1992)
M.F. Schilling, The longest run of heads. College Math. J. 21(3), 196–207 (1990)
R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141, 553–572 (1995)
V.H. Vu, New bounds on nearly perfect matchings of hypergraphs: higher codegrees do help. Random Struct. Algorithms 17, 29–63 (2000)
V.H. Vu, Concentration of non-Lipschitz functions and applications. Random Struct. Algorithms 20(3), 262–316 (2002)
A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)
Y. Zhao, Constructing MSTD sets using bidirectional ballot sequences. J. Number Theory 130(5), 1212–1220 (2010)
Y. Zhao, Sets characterized by missing sums and differences. J. Number Theory 131, 2107–2134 (2011)
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Iyer, G., Lazarev, O., Miller, S.J., Zhang, L. (2014). Finding and Counting MSTD Sets. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_7
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