On sums of subsets of a set of integers. (English) Zbl 0666.10035
For each set \(A\subset \{1,2,\ldots,n\}\), let \(A^*=\{\sum_{b\in B}b: B\subseteq A\}\). Let \(p(n,r)\), \(r\geq 2\) (resp. \(f(n,m)\), \(m\geq 1\)) be such that \(A^*\) contains no \(r\)th power of an integer (resp. such that \(m\not\in A^*)\). Estimates on asymptotic formulae for \(p(n,r)\) and \(f(n,m)\) are obtained; a special case gives a conjecture of Erdős and Graham that \(f(n,m)\sim (\tfrac 12+o(1))(n/\log n)\). The proofs rely on a technical result which is established using the Hardy-Littlewood method.
Reviewer: M. M. Dodson (Heslington)
MSC:
11B13 | Additive bases, including sumsets |
11P55 | Applications of the Hardy-Littlewood method |
11B75 | Other combinatorial number theory |
05A05 | Permutations, words, matrices |
Citations:
Zbl 0622.10042References:
[1] | N. Alon, Subset sums,J. Number Theory,27 (1987), 196–205. · Zbl 0622.10042 · doi:10.1016/0022-314X(87)90061-8 |
[2] | P.Erdos and G.Freiman, On two additive problems,J. Number Theory, to appear. |
[3] | P.Erdos, Some problems and results on combinatorial number theory,Proc. First China Conference in Combinatorics (1986),to appear. · Zbl 0609.47053 |
[4] | E.Lipkin, On representation ofr-powers by subset sums,Acta Arithmetica, submitted. · Zbl 0691.10042 |
[5] | J. Olson, An additive theorem modulop, J. Combinatorial Theory 5 (1968), 45–52. · Zbl 0174.05202 · doi:10.1016/S0021-9800(68)80027-4 |
[6] | E. C. Titchmarsh,Introduction to the theory of Fourier integrals, Oxford at the Clarendon Press, London, 1948, p. 177. · Zbl 0031.03202 |
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