---
title: Buck's Measure Density
url: https://www.emergentmind.com/topics/buck-s-measure-density
type: topic
---

# Buck's Measure Density

Searching arXiv for recent and foundational papers on Buck's measure density and related results.
Buck’s measure density is a measure-theoretic notion of density on subsets of the natural numbers generated by finite coverings by arithmetic progressions. In its classical form, for \(S\subset \mathbb N\) one defines
\[
\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},
\]
where \(r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}\). A set is Buck measurable when
\[
\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.
\]
This construction, introduced by R. C. Buck in 1946 and developed in later work, lies at the intersection of arithmetic progressions, modular distribution, outer-measure constructions, and finitely additive probability on suitable algebras of sets [2508.00538], [1502.01611].

## 1. Classical definition and measurable structure

In the classical notation, \((m)=0+(m)\) is the set of multiples of \(m\), and for \(S\subset \mathbb N\), \(aS=\{as;\ s\in S\}\). Buck’s outer density \(\mu^\ast\) is defined by minimizing the total cost \(\sum 1/m_i\) over finite coverings of \(S\) by residue classes \(r_i+(m_i)\). The family of Buck measurable sets is denoted by \(\mathcal D_\mu\), and it satisfies two basic structural properties: \(\mathcal D_\mu\) is an algebra of sets, and the restriction
\[
\mu=\mu^\ast|_{\mathcal D_\mu}
\]
is a finitely additive probability measure on \(\mathcal D_\mu\) [2508.00538].

This is the direct arithmetic analogue of an outer-measure-to-measure construction. In later terminology, the upper Buck density is often written \(b^\star\) or \(\mathfrak b^\star\), the conjugate lower Buck density as \(b_\star\) or \(\mathfrak b_\star\), and the induced Buck density \(b\) is defined on the domain
\[
\operatorname{dom}(b)=\{X\subseteq \mathbf N: b_\star(X)=b^\star(X)\},
\]
with
\[
b(X)=b^\star(X)=b_\star(X)
\]
on that domain [2212.11176]. This distinction is fundamental: upper and lower Buck densities are defined on all subsets, whereas Buck density in the strict measurable sense is defined only where the upper and lower values coincide [1510.07473].

## 2. Residue-class formulations and arithmetic meaning

A major structural feature of Buck density is that it admits exact reformulations in terms of residue classes modulo highly divisible integers. For \(S\subset\mathbb N\) and \(m\in\mathbb N\), let
\[
R(S:m)=\bigl|\{s \pmod m;\ s\in S\}\bigr|,
\]
the number of residue classes modulo \(m\) hit by \(S\). If \(\{B_N\}\) is a sequence such that every fixed \(d\in\mathbb N\) divides all sufficiently large \(B_N\), then
\[
\mu^\ast(S)=\lim_{N\to\infty}\frac{R(S:B_N)}{B_N}
\]
for every \(S\subset\mathbb N\) [2508.00538]. In this form, Buck’s outer density is literally the limiting proportion of occupied residue classes modulo sufficiently divisible moduli.

A parallel formulation in additive-combinatorial language uses residue images \(X^{(m)}\) modulo \(m\). For any multiplicatively increasing exhaustive sequence \((m_n)\),
\[
\mathfrak b^\star(X)=\lim_{n\to\infty}\frac{|X^{(m_n)}|}{m_n}
\quad\text{and}\quad
\mathfrak b_\star(X)=\lim_{n\to\infty}\frac{|X_\ast^{(m_n)}|}{m_n},
\]
so upper Buck density is asymptotically the proportion of residue classes modulo \(m_n\) that intersect \(X\) [2410.13275].

This residue-class viewpoint explains both the power and the counterintuitive behavior of Buck density. It is not a sparsity notion based on counting points in intervals. Leonetti and Tringali exhibit the set
\[
X:=\{h!+h:h\in\mathbf N\}
\]
and state that \(\mathfrak b^\ast_{\mathbf H}(X)=1\) [1905.08075]. Likewise, in additive-combinatorial examples built from irrational rotations, one has \(u^\star(A_\alpha)=\alpha\) but \(\mathfrak b^\star(A_\alpha)=1\) and \(\mathfrak b_\star(A_\alpha)=0\) [2410.13275]. These examples show that Buck density measures modular spread rather than interval frequency.

A further interpretation, noted in [2508.00538], is
\[
\mu^\ast(S)=P(\operatorname{cl}(S)),
\]
where closure is taken in the compact ring of polyadic integers and \(P\) is Haar measure. This suggests a topological-measure realization of Buck density as a Haar-measure shadow of closure in a profinite-type compactification.

## 3. Explicit constructions and computable examples

The constructive side of the theory is especially clear in the recent paper “On Buck’s measurability of certain sets” [2508.00538]. A key technical tool is a weak \(\sigma\)-additivity principle: if \(A_1,A_2,\dots\) are disjoint sets in \(\mathcal D_\mu\) and
\[
\lim_{N\to\infty}\mu^\ast\Big(\bigcup_{k=N}^\infty A_k\Big)=0,
\]
then
\[
A=\bigcup_{k=1}^\infty A_k\in \mathcal D_\mu
\quad\text{and}\quad
\mu(A)=\sum_{k=1}^\infty \mu(A_k).
\]
This substitutes for countable additivity in situations where the tails become Buck-negligible.

A second recurring principle is dilation. If \(b_i\) is an increasing sequence with \(b_i\mid b_{i+1}\), and \(H_i\in\mathcal D_\mu\) are such that all elements of the union of these sets are relatively prime to all \(b_i\), then
\[
H=\bigcup_{i=1}^\infty b_iH_i
\]
is Buck measurable and
\[
\mu(H)=\sum_{i=1}^\infty \frac{\mu(H_i)}{b_i}.
\]
Under the hypotheses used in the paper, the scaling law is
\[
\mu(b_iH_i)=\frac{\mu(H_i)}{b_i}
\]
[2508.00538].

These principles yield explicit classes of measurable sets with computable Buck measures.

| Set | Buck measure |
|---|---|
| \(\mathbf O\), the odd numbers | \(\mu(\mathbf O)=\frac12\) |
| \(B_\alpha\), for \(\alpha\in[0,1]\) | \(\mu(B_\alpha)=\alpha\) |
| \(N(p,E)\), with \(E=\{e_1<e_2<\cdots\}\) | \(\mu(N(p,E))=\Bigl(1-\frac1p\Bigr)\sum_{n=1}^\infty \frac1{p^{e_n}}\) |
| \(N(p_1,\dots,p_k,E_1,\dots,E_k)\) | \(\mu(N)= \prod_{i=1}^k\Bigl(1-\frac1{p_i}\Bigr) \prod_{i=1}^k\sum_{n_i\in E_i}\frac1{p_i^{n_i}}\) |

The set \(N(p,E)\) consists of integers whose \(p\)-adic exponent belongs to the prescribed set \(E\), and it decomposes as
\[
N(p,E)=\bigcup_{n=1}^\infty p^{e_n}(\mathbb N\setminus (p)).
\]
The several-prime version imposes simultaneous exponent restrictions on finitely many prime coordinates in the canonical factorization [2508.00538].

These examples show that Buck density interacts naturally with multiplicative structure. The measurable sets are not limited to periodic sets themselves; they include infinite disjoint unions of suitably scaled measurable pieces and sets defined by conditions on prime exponents.

## 4. Nullity criteria and arithmetic zero sets

The second major theme is Buck-nullity. For a prime \(p\) and \(S\subset\mathbb N\), define
\[
S_p=\{s\in S;\ p\mid s \ \land\ p^2\nmid s\},
\]
the slice of elements of \(S\) with \(p\)-adic valuation exactly \(1\). In the Buck setting, Niven’s theorem is stated as follows: if \(\{p_i\}\) is a sequence of primes with
\[
\sum_{i=1}^\infty \frac1{p_i}=\infty,
\]
then
\[
\mu(S)=0 \iff \forall i=1,2,\dots,\ \mu(S_{p_i})=0
\]
[2508.00538]. This reduces the proof of Buck-nullity to the analysis of first-layer prime slices.

The paper applies this criterion to several concrete families. If \(S\) satisfies
\[
\forall s\in S\ \forall n,\quad p_n\mid s \Rightarrow p_n^2\mid s,
\]
then \(\mu(S)=0\). If \(R_t\) denotes the set of integers containing at most \(t\) selected primes with odd exponent in canonical representation, then
\[
R_t\in\mathcal D_\mu,\qquad \mu(R_t)=0.
\]
If \(P_t\) is the set of integers containing at most \(t\) primes in canonical representation, then
\[
P_t\in\mathcal D_\mu,\qquad \mu(P_t)=0.
\]
Finally, for
\[
R=\{n\in\mathbb N;\ \tau(n)\mid n\},
\]
where
\[
\tau(n)=(\alpha_1+1)\cdots(\alpha_k+1)
\quad\text{for}\quad
n=p_1^{\alpha_1}\cdots p_k^{\alpha_k},
\]
the paper proves
\[
R\in\mathcal D_\mu,\qquad \mu(R)=0
\]
[2508.00538].

A broader structural theorem is due to Leonetti and Tringali: a set \(X\subseteq \mathbf H\) is small, meaning \(\mu^\ast(X)=0\) for every upper quasi-density \(\mu^\ast\), if and only if
\[
\mathfrak b^\ast_{\mathbf H}(X)=0,
\]
equivalently if and only if \(\mathfrak b^\ast_{\mathbf Z}(X)=0\) [1905.08075]. In this precise sense, upper Buck density is the universal detector of null sets for the whole class of upper quasi-densities considered there. The same paper derives Buck-nullity for integers with less than a fixed number of prime factors, for values of a binary quadratic form whose discriminant is not a perfect square, and for the image of \(\mathbf Z\) through a non-linear integral polynomial in one variable [1905.08075].

## 5. Axiomatic placement and generalized extensions

Buck density now sits inside a general axiomatic theory of densities on \(\mathcal P(\mathbb N)\). In the framework of Leonetti and Tringali, an upper density \(\mu^\star\) satisfies normalization, monotonicity, subadditivity, and the translation-dilation law
\[
\mu^\star(k\cdot X+h)=\frac1k\mu^\star(X)
\qquad
(X\subseteq H,\ h,k\in\mathbb N^+),
\]
and upper Buck density is explicitly listed among the standard examples [1510.07473]. Their main theorem states that every upper quasi-density has the strong Darboux property. Consequently, upper Buck density and lower Buck density satisfy the interpolation principle
\[
\forall X\subseteq Y\subseteq \mathbb N,\ \forall a\in[f(X),f(Y)],\ \exists A\text{ with }X\subseteq A\subseteq Y\text{ and }f(A)=a,
\]
for \(f=\mathfrak b^\star\) and likewise for \(f=\mathfrak b_\star\) [1510.07473]. This strengthens mere surjectivity onto \([0,1]\): every intermediate Buck-density value is realizable inside every inclusion interval of sets.

A different generalization replaces arithmetic progressions by an abstract system of finite decompositions. Given a set \(X\), finite decompositions
\[
\mathcal E_n =\{A_1^{(n)}, \dots,A_{k_n}^{(n)} \},
\]
satisfying a common-refinement condition and a point-separation condition, and a finitely additive probability measure \(\Delta\) on the algebra \(\mathcal D_0\) generated by the decomposition atoms, one defines
\[
\nu^*(S) = \inf \{\Delta(A); A \in \mathcal D_0,\ S \subset A \}.
\]
A set is \(\nu^*\)-measurable iff
\[
\nu^*(S)+\nu^*(X\setminus S)=1,
\]
and the restriction \(\nu=\nu^*|_{\mathcal D_\nu}\) is a finitely additive probability measure [1502.01611]. The classical Buck density is recovered by taking \(X=\mathbb N\), \(\mathcal E_n\) equal to the residue classes modulo \(n!\), and \(\Delta(A_j^{(n)})=1/n!\). In this generalized form, Buck’s construction becomes a prototype for a decomposition-based outer density linked to compact metric completions, Borel probability measures, and uniform distribution theory [1502.01611].

## 6. Universal comparison principles and additive-combinatorial role

Buck density has acquired a central role well beyond its original measure-theoretic setting. In “On the density of sumsets, II”, the decisive comparison principle is
\[
b_\star(X) \le \mu_\star(X) \le \mu^\star(X) \le b^\star(X)
\qquad \text{for every } X\subseteq \mathbf N,
\]
for every arithmetic quasi-density \(\mu\) [2212.11176]. Moreover,
\[
\mathcal A \subseteq \operatorname{dom}(b) \subseteq \operatorname{dom}(\mu)
\quad\text{and}\quad
\mu(X)=b(X)\qquad \text{for every } X\in \operatorname{dom}(b).
\]
Thus Buck density acts as a universal comparison object, and on the Buck-measurable domain every arithmetic quasi-density agrees with it.

The same paper identifies the practical nullity criterion
\[
b(X)=0 \iff X \text{ covers } o(n!) \text{ residue classes modulo } n! \text{ as } n\to\infty
\]
and uses it to prove that if \(B\subseteq \mathbf N\) is non-empty and \(b(B)=0\), then for each \(\alpha\in[0,1]\) there exists \(A\subseteq \mathbf N\) such that, for every arithmetic quasi-density \(\mu\), both \(A\) and \(A+B\) are in the domain of \(\mu\) and
\[
\mu(A + B) = \alpha
\]
[2212.11176]. This relies on periodic approximants modulo \(n!\) and on the fact that Buck-null sets occupy only asymptotically negligible numbers of residue classes.

In additive combinatorics, upper Buck density supports a Kneser-type inverse theorem. If \(k\ge 2\) and
\[
\mathfrak b^\star(X_1+\cdots+X_k) < \sum_{i=1}^k \mathfrak b^\star(X_i),
\]
then there exist a positive integer
\[
q\le \frac{2k-2}{\eta\sigma},
\]
periodic sets \(A_i\), and exact finite-quotient formulas such as
\[
\mathfrak b^\star(X_1+\cdots+X_k)
=
\frac{\sum_{i=1}^k(r_i-1)+1}{q},
\]
together with a quasi-periodic or arithmetic-progression alternative in \(\mathbb Z/q\mathbb Z\) [2410.13275]. This is the upper-Buck analogue of the “small sumset implies periodic structure” paradigm.

The same line of work also exhibits markedly nonclassical behavior. For any \(\gamma\in[0,1)\), there exists \(A\subseteq\mathbb N\) such that
\[
\mathfrak b(A)=0,\qquad \mathfrak b_\star(A+A)=0,\qquad \mathfrak b^\star(A+A)=\gamma,
\]
and for \(0\le \alpha<\tfrac14\) there exists \(A\subseteq\mathbb N\) such that
\[
\mathfrak b(A)=\alpha
\quad\text{and}\quad
\mathfrak b_\star(A+A)<\mathfrak b^\star(A+A),
\]
so \(A\) has a Buck density while \(A+A\) does not [2410.13275]. These constructions reinforce a general point: Buck density is highly arithmetic, governed by congruence-class occupancy across scales rather than by interval growth alone.

Source: https://www.emergentmind.com/topics/buck-s-measure-density