On a problem of Sárközy
Let \(A,B\subseteq \mathbb N\). The set addition of \(A\) and \(B\) is defined by \[ A+B:=\{a+b\mid a\in A,\;b\in B\} . \] For \(A\subseteq \mathbb N\) the lower density of \(A\) is \[ \underline{d}(A):=\liminf_{n\to \infty} \frac{ | A\cap [1,n]| }{n}. \] and the upper density \[ \bar d(A):=\limsup_{n\to \infty} \frac{| A\cap [1,n]| }{n}, \] Sárközy presented the following conjecture: For infinite sets \(A,B\subseteq \mathbb N\) with positive lower densities and irrational number \(\alpha\) \[ a^2+b^2= \lfloor n\alpha \rfloor \quad (a\in A,\;b\in B ) \] has infinitely many solutions. One of the results of this note is the following theorem: Let \(A\subseteq \mathbb N\) be an infinite sequence with \(\bar d(A)>\frac{1}{2} -\frac{1}{2a}\) and \(p(x)\) a polynomial with positive integer coefficients. Then \[ | \{p(a_1)+p(a_2):a_i\in A,\;i=1,2\}\cap N(\alpha)| =\infty, \] where \(N(\alpha)=\{\lfloor n\alpha\rfloor :n\in \mathbb N\}\).
scientific article; zbMATH DE number 2108667