On the size of primitive sets in function fields (Q1987098)

Let \(\mathcal{M}_q\) denote the set of monic polynomials in \(\mathbb{F}_q[x]\) and \(\mathcal{I}_q\) denote the set of irreducible polynomials in \(\mathcal{M}_q\). Just as in the integers, we say that a set \(A\subset\mathcal{M}_q\) is primitive if no element divides another. The aim of the paper under review is to obtain some results concerning primitive sets in the function field analogous to those in integers. The authors show that the lower density of a primitive set in the function field is always zero. Then, motivated by studying primitive sets with optimal upper density, they give a construction of a set with upper density arbitrarily close to \(1-1/q\). Finally, they prove the existence of primitive sets \(S\subset\mathcal{M}_q\) with consistently growing counting functions \(S'(n)=|\{f \in S : \deg(f) = n\}|\) of size \[ S'(n)\asymp\frac{q^n}{\left(\log n\right)L(\log n)}, \] where \(L(x)=\log_2 x\,\log_3 x\cdots\left(\log_{j-1} x\right)^{1+\varepsilon}\) (for \(j\geq 2\)) and \(\log_k x\) denotes the \(k\)-fold iterated logarithm.