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Sets of integers with missing differences. (English) Zbl 0956.11006

Given a set of positive integers \(M\), a set \(S\) of positive integers is said to be an \(M\)-set if \(a,b\in S\) implies \(a-b\notin M\). If \(\mu(M)\) denotes the supremum of the upper (asymptotic) densities of the all \(M\)-sets, and \(d(M)=\sup_{x\in(0,1)} \min_{m\in M}\|xm\|\), then D. G. Cantor and B. Gordon [J. Comb. Theory, Ser. A 14, 281-287 (1973; Zbl 0277.10043)] proved that \(\mu(M)\geq d(M)\). The author proves
(1) the exact value for \(\mu(M)\) when \(M\) is a finite arithmetical sequence with a coprime difference and offset, or a four element set of the type \(M=\{i,j,2i,i+j\}\) with \((i,j)=1\) and \(j\equiv i+1\pmod{3}\), or \(M=\{i,j,i+j,2j\}\) with \((i,j)=1\) and \(j\equiv i+2\pmod{3}\),
(2) lower bounds for \(\mu(M)\) and \(d(M)\) for wide classes of three element sets \(M\), where in some cases the bounds are exact (some of them are conjectured to be exact).

MSC:

11B05 Density, gaps, topology
11B25 Arithmetic progressions

Citations:

Zbl 0277.10043
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References:

[1] Cantor, D. G.; Gordon, B., Sequences of integers with missing differences, J. Combin. Theory Ser. A, 14, 281-287 (1973) · Zbl 0277.10043
[2] Gupta, S., Some Results on the Dispersion Spectrum and Maximal Density of \(M\)-Sets (1997), Indian Institute of TechnologyDepartment of Mathematics: Indian Institute of TechnologyDepartment of Mathematics New Delhi
[3] Haralambis, N. M., Sets of integers with missing differences, J. Combin. Theory Ser. A, 23, 22-33 (1977) · Zbl 0359.10047
[4] T. S. Motzkin, Unpublished problem collection.; T. S. Motzkin, Unpublished problem collection.
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