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On certain solutions of the diophantine equation \(x-y=p(z)\). (English) Zbl 0776.11006

A subset \(A\) of \(\mathbb{N}\) is called intersective if for each subset \(S\) of \(\mathbb{N}\) with positive Banach density \(b(S)>0\) the set \(A\cap (S-S)\) is non-empty. Equivalently \(A\) is a set of recurrence [A. Bertrand- Mathis, Isr. J. Math. 55, 184-198 (1986; Zbl 0611.10032)].
Using ergodic theory and the uniform distribution of polynomial values of sequences of primes in certain arithmetic progressions it is shown that for a polynomial \(\psi\) with integer coefficients the set \(P_ \psi=\{\psi(p)\), \(p\) a prime number} is intersective if and only if for each non-zero integer \(n\), there exists an integer \(m_ n\) coprime to it, such that \(n\) divides \(\psi(m_ n)\). The case \(\psi(x)=x-1\) was first considered by A. Sarközy [Acta Math. Acad. Sci. Hung. 31, 355-386 (1978; Zbl 0387.10034)].
Furthermore the author considers countable commutative monoids \(M\), and collections of subsets \({\mathcal A}=\{A_ n\}_{n=1}^ \infty\) satisfying some Følner like conditions (in condition (iv) the first “\(A_ n\)” should be omitted) and proves some results in the spirit of V. Bergelson’s paper [J. Lond. Math. Soc., II. Ser. 31, 295-304 (1985; Zbl 0579.10029)].
Reviewer: H.Rindler (Wien)

MSC:

11B05 Density, gaps, topology
11K06 General theory of distribution modulo \(1\)
28D99 Measure-theoretic ergodic theory
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