论文信息 - Sumsets Contained in Infinite Sets of Integers

Sumsets Contained in Infinite Sets of Integers

Abstract If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. Erdos conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ⊆ A. This conjecture is still unproved. This paper contains several results on sumsets contained in finite sets of integers. For example, if A is a set of integers of positive upper density, then for any n there exist sets B and F such that B has positive upper density, F has cardinality n, and B + F ⊆ A.

Melvyn B Nathanson

[1] Paul Erdös,et al. On Some Sequences of Integers , 1936 .

[2] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions , 1977 .

[3] R. Graham,et al. Ramsey’s theorem for $n$-parameter sets , 1971 .

[4] James E. Baumgartner,et al. A Short Proof of Hindman's Theorem , 1974, J. Comb. Theory, Ser. A.

[5] A. Khinchin. Three Pearls of Number Theory , 1952 .

[6] Neil Hindman,et al. Finite Sums from Sequences Within Cells of a Partition of N , 1974, J. Comb. Theory, Ser. A.

[7] E. Szemerédi. On sets of integers containing k elements in arithmetic progression , 1975 .

[8] Neil Hindman. Partitions and Sums of Integers with Repetition , 1979, J. Comb. Theory, Ser. A.

[9] Neil Hindman. Partitions and sums and products of integers , 1979 .

[10] P. Erdös. Problems and results on combinatorial number theory III , 1977 .

[11] W. Comfort,et al. Ultrafilters: Some old and some new results , 1977 .