## Buck’s measure density and sets of positive integers containing arithmetic progression.(English)Zbl 0761.11004

{For notations and related results see the preceding review.} Let $$D_ 2$$ be the class of all $$S\subseteq\mathbb{N}$$ possessing a natural density $$d(S)$$, and let $$D_ 3$$ be the class of all “$$d$$-measurable” sets $$S\subseteq\mathbb{N}$$. The authors re-prove the result by Buck [loc. cit.] that $$D_ 3=D_ 2$$. Furthermore, they give another proof of assertion (i) above. Finally, they show the existence of a set $$S_ 0$$, with $$\mu^*(S_ 0)=1$$, not containing any three-term arithmetic progression.

### MSC:

 11B05 Density, gaps, topology 11B83 Special sequences and polynomials 28E99 Miscellaneous topics in measure theory

### Citations:

Zbl 0761.11003; Zbl 0061.075
Full Text:

### References:

 [1] BUCK R. C.: The measure theoretic approach to density. Amer. J. Math. LXVIII (1946), 560-580. · Zbl 0061.07503 [2] DINCULEANU N.: Vector Measures. VEB Deutscher Verlag der Wissen, Berlin, 1966 · Zbl 0647.60062 [3] ERDÖS P., NATHANSON M. B., SÁRKÖZY A.: Sumsets containing infinite arithmetic progressions. J. Number Theory 28 (1988), 159-166. · Zbl 0633.10047 [4] HARDY G. H., WRIGHT E. M.: An Introduction to the Theory of Numbers. 3rd, Oxford, 1954. · Zbl 0058.03301 [5] MAHARAM D.: Finitely additive measures on the integers. Sankhya: The Indian J, Stat. Ser. A 38 (1976), 44-59. · Zbl 0383.60008 [6] PAŠTÉKA M.: Some properties of Buck’s measure density. · Zbl 0761.11003 [7] SIERPIŃSKI W.: Elementary Theory of Numbers. PWN, Warszawa, 1964. · Zbl 0122.04402 [8] SIKORSKI R.: Funkcje rzeczywiste I. PWN, Warszawa, 1958. · Zbl 0093.05603 [9] SZEMEREDI E.: On sets of integers containing no k elements in arithmetic progression. Acta Arithm. 27 (1975), 199-245.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.