## 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

### Citations:

Zbl 0611.10032; Zbl 0387.10034; Zbl 0579.10029
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