Abstract
Let A be a subset of an abelian group. Let hA denote the set of all sums of h elements of A with repetitions allowed, and let h^A denote the set of all sums of h distinct elements of A, that is, all sums of the form a 1 + … + a h, where a 1, …, a h ∈ A and a i ≠ a j for i ≠ j.
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References
N. Alon, M, B. Nathanson, and I. Z. Ruzsa. Adding distinct congruence classes modulo a prime. Amer. Nlath. Monthly, 102, 1995.
W. Brakemeier. Eine Anzahlformel von Zahlen modulo n. Monat. Math., 85: 277–282, 1978.
A. L. Cauchy. Recherches sur les nombres. J. École poiutecli., 9: 99–116, 1813.
I Chowla. A theorem on the addition of residue classes: Application to the number Γ(k) in Waring’s problem. Proc. Indian Acad. Sci., Section .A, 1: 242–243, 1935.
H. Davenport. On the addition of residue classes. J. London Math. Soc., 10: 30–32, 1935.
H. Davenport. A historical note. J. London Math. Soc., 22: 100–101, 1947.
J. A. Dias da Silva and Y. O. Hamidoune. A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra and its Applications, 141: 283–287, 1990.
J. A. Dias da Silva and Y.a. Hamidoune. Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26: 140–146, 1994.
P. Erdôs and R. L. Graham. Old and New Problems and Resuite in Combinatorial Number Theory. L’Enseignement Mathématique, Geneva, 1980.
P. Erdôs and HL Heilbronn. On the addition of residue classes mod p. Acta Arith., 9: 149–159, 1964.
G. A. Freiman, L. Low, and J. Pitman. The proof of Paul Erdôs’ conjecture of the addition of different residue classes modulo prime number. In Structure Theory of Set Addition, 7–11 June 1993, CIRM Marseille, pages 99–108, 1993.
Y. O. Hamidoune. A generalization of an addition theorem of Shatrowsky. Europ. J. Combin., 13: 249–255, 1992.
R. Mansfield. How many slopes in a polygon? Israel J. Math., 39: 265–272, 1981.
S. S. Pillai. Generalization of a theorem of Davenport on the addition of residue classes. Proc. Indian Acad. Sci. (Series A), 6: 179–180, 1938.
J. M. Pollard. A generalization of a theorem of Cauchy and Davenport. J. London Math. Soc., 8: 460–462, 1974.
J. M. Pollard. Additive properties of residue classes. J. London Math. Soc., 11: 147–152, 1975.
L. Pyber. On the Erdôs-Heilbronn conjecture, personal communication.
U.-W. Rickert. Über eine Vermutung in der additiven Zahlentkeorie. PhD thesis, Tech. Univ. Braunschweig, 1976.
O. Rôdseth. Sums of distinct residues mod p. Acta Arith., 65: 181–184, 1993.
L. Shatrovskii. A new generalization of Davenport VPillai’s theorem on the addition of residue classes. Doklady Akad. Nauk CCCR, 45: 315–317, 1944.
R. Spigler. An application of group theory to matrices and to ordinary differential equations. Linear Algebra and its Applications, 44: 143–151, 1982.
D. Zeilberger. Andre’s reflection proof generalized to the many-candidate ballot problem. Discrete Math., 44: 325–326, 1983.
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Nathanson, M.B. (1997). Ballot Numbers, Alternating Products, and the Erdős-Heilbronn Conjecture. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös I. Algorithms and Combinatorics, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60408-9_16
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