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Sets with distinct sums of pairs, long arithmetic progressions, and continuous mappings. (English) Zbl 1424.43006

We say that \(t \in \mathbb{ R}\) is a point of nonlinearity of a continuous mapping \(\phi: \mathbb{R}\rightarrow \mathbb{ R}\) if \(t\) has no neighborhood in which \(\phi\) coincides with a linear function. The set \(E(\phi)\) of all such points is called the set of nonlinearity of \(\phi\).
The main result of this paper is the following Theorem. Let \(\phi:\mathbb{ R}\rightarrow \mathbb{ R}\) be continuous. Suppose that \(E(\phi)\) has nonzero Lebesgue measure. Then there exist a set \(A\) that contains arbitrarily long arithmetic progressions and a set \(B\) with distinct sums of pairs such that \(\phi\) maps bijectively \(A\) onto \(B\).

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
42A75 Classical almost periodic functions, mean periodic functions
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References:

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