Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence ( a ( n )) the sequence ( na ( n )) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset 𝒜𝒪𝒮 of ℓ _1 consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in 𝒜𝒪𝒮 and its subsets; this kind of study is known as lineability. Finally we show that 𝒜𝒪𝒮 is a residual 𝒢_δσ but not an ℱ_σδ-set .