Homotopy transfer and formality

In this paper, we prove that, given appropriate hypotheses, n-formality of a differential graded algebraic structure is equivalent to the existence chain-level automorphism lifting degree twisting isomorphism relative unit order greater than n. A similar result with slightly different hypothesis was proved by Joana Cirici and second author. We use homotopy transfer theorem an explicit inductive procedure in kill higher operations. As application our result, formality coefficients p-adic integers certain dg-algebras coming from hyperplane toric arrangements configuration spaces.