### Abstract

If a finite set A of integers included in {1 , . . . , N} has more than N/k elements, one may expect that the set ℓA of sums of ℓ elements of A, contains, when ℓ is comparable to k, a rather long arithmetic progression (which can be required to be homogeneous or not). After presenting the state of the art, we show that some of the results cannot be improved as far as it would be thought possible in view of the known results in the infinite case. The paper ends with lower and upper bounds for the order, as asymptotic bases, of the subsequences of the primes which have a positive relative density.

Original language | English |
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Pages (from-to) | 109-127 |

Number of pages | 19 |

Journal | Asterisque |

Volume | 258 |

Publication status | Published - 1999 |

### Fingerprint

### Keywords

- Additive bases
- Additive number theory
- Density
- Structure theory of set addition

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On finite addition theorems.** / Sárközy, A.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On finite addition theorems

AU - Sárközy, A.

PY - 1999

Y1 - 1999

N2 - If a finite set A of integers included in {1 , . . . , N} has more than N/k elements, one may expect that the set ℓA of sums of ℓ elements of A, contains, when ℓ is comparable to k, a rather long arithmetic progression (which can be required to be homogeneous or not). After presenting the state of the art, we show that some of the results cannot be improved as far as it would be thought possible in view of the known results in the infinite case. The paper ends with lower and upper bounds for the order, as asymptotic bases, of the subsequences of the primes which have a positive relative density.

AB - If a finite set A of integers included in {1 , . . . , N} has more than N/k elements, one may expect that the set ℓA of sums of ℓ elements of A, contains, when ℓ is comparable to k, a rather long arithmetic progression (which can be required to be homogeneous or not). After presenting the state of the art, we show that some of the results cannot be improved as far as it would be thought possible in view of the known results in the infinite case. The paper ends with lower and upper bounds for the order, as asymptotic bases, of the subsequences of the primes which have a positive relative density.

KW - Additive bases

KW - Additive number theory

KW - Density

KW - Structure theory of set addition

UR - http://www.scopus.com/inward/record.url?scp=33748580319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748580319&partnerID=8YFLogxK

M3 - Article

VL - 258

SP - 109

EP - 127

JO - Asterisque

JF - Asterisque

SN - 0303-1179

ER -