## Abstract

Let k be a field, G be an abelian group and r∈ N. Let L be an infinite dimensional k-vector space. For any m∈ End _{k}(L) we denote by r(m) ∈ [ 0 , ∞] the rank of m. We define by R(G, r, k) ∈ [ 0 , ∞] the minimal R such that for any map A: G→ End _{k}(L) with r(A(g^{′}+ g^{′ ′}) - A(g^{′}) - A(g^{′ ′})) ≤ r, g^{′}, g^{′ ′}∈ G there exists a homomorphism χ: G→ End _{k}(L) such that r(A(g) - χ(g)) ≤ R(G, r, k) for all g∈ G. We show the finiteness of R(G, r, k) for the case when k is a finite field, G= V is a k-vector space V of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of Approximate Cohomology groups HFk(V,M) [which is a purely algebraic analogue of the notion of ϵ-representation (Kazhdan in Isr. J. Math. 43:315–323, 1982)] and interperate our result as a computation of the group HF1(V,M) for some V-modules M.

Original language | American English |
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Pages (from-to) | 499-509 |

Number of pages | 11 |

Journal | Selecta Mathematica, New Series |

Volume | 24 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2018 |

### Bibliographical note

Publisher Copyright:© 2017, Springer International Publishing.

## Keywords

- Algebraic rank
- Bias
- Cohomology
- Exponential sums
- Gowers norms