On Small Sets of Integers

An upper quasi-density on $\mathbf{H}$ (the integers or the non-negative integers) is a real-valued subadditive function ${\mu }^{\ast }$ defined on the whole power set of $\mathbf{H}$ such that ${\mu }^{\ast }(X)\le {\mu }^{\ast }(\mathbf{H})=1$ and ${\mu }^{\ast }(k\cdot X+h)=\frac{1}{k}{\mu }^{\ast }(X)$ for all $X \subseteq \bf H$,$k \in {\bf N}^+$,and$h \in \bf N$,where$k \cdot X := {kx: x \in X}$; and an upper density on $\mathbf{H}$ is an upper quasi-density on $\mathbf{H}$ that is non-decreasing with respect to inclusion. We say that a set $X\subseteq \mathbf{H}$ is small if ${\mu }^{\ast }(X)=0$ for every upper quasi-density ${\mu }^{\ast }$ on $\bf H$. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper P'olya densities, along with the uncountable family of upper $\alpha$-densities, where $\alpha$ is a real parameter $\ge -1$ (most notably, $\alpha =-1$ corresponds to the upper logarithmic density, and $\alpha =0$ to the upper asymptotic density). It turns out that a subset of $\mathbf{H}$ is small if and only if it belongs to the zero set of the upper Buck density on $\mathbf{Z}$. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of $\mathbf{Z}$ through a non-linear integral polynomial in one variable.