A remark on a result of Helfgott, Roton and Naslund

Abstract

Let $F(X)={\prod }_{i=1}^k(a_{i}X+b_i)$ be a polynomial with $a_i,b_i$ being integers. Suppose the discriminant of $F$ is non-zero and $F$ is admissible. Given any natural number $N$, let $S(F,N)$ denotes those integers less than or equal to $N$ such that $F(n)$ has no prime factors less than or equal to $N^{1/(4k+1)}.$ Let $L$ be a translation invariant linear equation in $3$ variables. Then any $A\subset S(F,N)$ with ${\delta }_F(N):=\frac{|A|}{|S(F,N)|}{\gg }_{\epsilon ,F,L}\frac{1}{(\log \log N{)}^{1-\epsilon }}$ contains a non-trivial solution of $L$ provided $N$ is sufficiently large.