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On Small Sets of Integers

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $μ^\ast$ defined on the whole power set of $\mathbf H$ such that $μ^\ast(X) \le μ^\ast({\bf H}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k}\, μ^\ast(X)$ 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 $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $μ^\ast(X) = 0$ for every upper quasi-density $μ^\ast$ on $\bf H$. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper $α$-densities, where $α$ is a real parameter $\ge -1$ (most notably, $α= -1$ corresponds to the upper logarithmic density, and $α= 0$ to the upper asymptotic density). It turns out that a subset of $\bf H$ is small if and only if it belongs to the zero set of the upper Buck density on $\bf Z$. 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 $\bf Z$ through a non-linear integral polynomial in one variable.