Addition of sets of integers of positive density
A theorem of Freiman describes (qualitatively) the structure of finite sets satisfying \( |A+A |\leq \sigma |A |\) (with constant \(\sigma \)) as subsets of a certain generalized arithmetic progression. Freiman gave also an extension to infinite sets, where the condition is replaced by \( \underline d(A+A) \leq \sigma \underline d(A)\), under the additional constraint that the elements of the set arranged in increasing order satisfy \( \limsup a_{i+1}/a_i \leq \Lambda \) with another constant \(\Lambda \). The covering set is a ``Bohr set'' of the form \[ {\mathcal B } = \{n: n\alpha _i \in (-\varepsilon _i/2, \varepsilon _i/2) \text{ mod } 1 \} \] with certain real numbers \(\alpha _1, ..., \alpha _m\) and \(0<\varepsilon _i<1\). Here the ``dimension'' satisfies \(m\leq [2\sigma -2]\) and the size \(\varepsilon _1 ... \varepsilon _m\) of \(\mathcal B\) is bounded by \(c(\sigma , \Lambda ) \underline d(A)\). In the present work this description is refined by showing that the Bohr set can be replaced by the direct product of a Bohr set of dimension \( \leq [ \log _2 \sigma ]\) and a generalized arithmetic progression of a rather special type.
scientific article; zbMATH DE number 1025610
scientific article