Buck's Measure Density

• Buck's measure density is defined via finite coverings by arithmetic progressions, measuring the modular spread of subsets of natural numbers.
• It distinguishes between upper, lower, and strict Buck densities, providing a robust framework for nullity criteria and finitely additive probability measures.
• The approach enables explicit computable examples and supports universal comparison principles in additive combinatorics and density theory.

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 (Pasteka, 1 Aug 2025, Iacò et al., 2015).

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 $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$0 by residue classes $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$1. The family of Buck measurable sets is denoted by $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$2, and it satisfies two basic structural properties: $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$3 is an algebra of sets, and the restriction

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$4

is a finitely additive probability measure on $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$5 (Pasteka, 1 Aug 2025).

This is the direct arithmetic analogue of an outer-measure-to-measure construction. In later terminology, the upper Buck density is often written $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$6 or $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$7, the conjugate lower Buck density as $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$8 or $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$9, and the induced Buck density $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$0 is defined on the domain

$r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$1

with

$r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$2

on that domain (Leonetti et al., 2022). 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 (Leonetti et al., 2015).

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 $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$3 and $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$4, let

$r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$5

the number of residue classes modulo $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$6 hit by $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$7. If $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$8 is a sequence such that every fixed $r+(m)=\{n\in\mathbb N;\ n\equiv r \pmod m\}$9 divides all sufficiently large $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$0, then

$\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$1

for every $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$2 (Pasteka, 1 Aug 2025). 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 $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$3 modulo $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$4. For any multiplicatively increasing exhaustive sequence $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$5,

$\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$6

so upper Buck density is asymptotically the proportion of residue classes modulo $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$7 that intersect $\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$8 (Hennecart, 2024).

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

$\mu^\ast(S)+\mu^\ast(\mathbb N\setminus S)=1.$9

and state that $(m)=0+(m)$0 (Leonetti et al., 2019). Likewise, in additive-combinatorial examples built from irrational rotations, one has $(m)=0+(m)$1 but $(m)=0+(m)$2 and $(m)=0+(m)$3 (Hennecart, 2024). These examples show that Buck density measures modular spread rather than interval frequency.

A further interpretation, noted in (Pasteka, 1 Aug 2025), is

$(m)=0+(m)$4

where closure is taken in the compact ring of polyadic integers and $(m)=0+(m)$5 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 paper “On Buck’s measurability of certain sets” (Pasteka, 1 Aug 2025). A key technical tool is a weak $(m)=0+(m)$6-additivity principle: if $(m)=0+(m)$7 are disjoint sets in $(m)=0+(m)$8 and

$(m)=0+(m)$9

then

$m$0

This substitutes for countable additivity in situations where the tails become Buck-negligible.

A second recurring principle is dilation. If $m$1 is an increasing sequence with $m$2, and $m$3 are such that all elements of the union of these sets are relatively prime to all $m$4, then

$m$5

is Buck measurable and

$m$6

Under the hypotheses used in the paper, the scaling law is

$m$7

(Pasteka, 1 Aug 2025).

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

Set	Buck measure
$m$8, the odd numbers	$m$9
$S\subset \mathbb N$0, for $S\subset \mathbb N$1	$S\subset \mathbb N$2
$S\subset \mathbb N$3, with $S\subset \mathbb N$4	$S\subset \mathbb N$5
$S\subset \mathbb N$6	$S\subset \mathbb N$7

The set $S\subset \mathbb N$8 consists of integers whose $S\subset \mathbb N$9-adic exponent belongs to the prescribed set $aS=\{as;\ s\in S\}$0, and it decomposes as

$aS=\{as;\ s\in S\}$1

The several-prime version imposes simultaneous exponent restrictions on finitely many prime coordinates in the canonical factorization (Pasteka, 1 Aug 2025).

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 $aS=\{as;\ s\in S\}$2 and $aS=\{as;\ s\in S\}$3, define

$aS=\{as;\ s\in S\}$4

the slice of elements of $aS=\{as;\ s\in S\}$5 with $aS=\{as;\ s\in S\}$6-adic valuation exactly $aS=\{as;\ s\in S\}$7. In the Buck setting, Niven’s theorem is stated as follows: if $aS=\{as;\ s\in S\}$8 is a sequence of primes with

$aS=\{as;\ s\in S\}$9

then

$\mu^\ast$0

(Pasteka, 1 Aug 2025). 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 $\mu^\ast$1 satisfies

$\mu^\ast$2

then $\mu^\ast$3. If $\mu^\ast$4 denotes the set of integers containing at most $\mu^\ast$5 selected primes with odd exponent in canonical representation, then

$\mu^\ast$6

If $\mu^\ast$7 is the set of integers containing at most $\mu^\ast$8 primes in canonical representation, then

$\mu^\ast$9

Finally, for

$\sum 1/m_i$0

where

$\sum 1/m_i$1

the paper proves

$\sum 1/m_i$2

(Pasteka, 1 Aug 2025).

A broader structural theorem is due to Leonetti and Tringali: a set $\sum 1/m_i$3 is small, meaning $\sum 1/m_i$4 for every upper quasi-density $\sum 1/m_i$5, if and only if

$\sum 1/m_i$6

equivalently if and only if $\sum 1/m_i$7 (Leonetti et al., 2019). 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 $\sum 1/m_i$8 through a non-linear integral polynomial in one variable (Leonetti et al., 2019).

5. Axiomatic placement and generalized extensions

Buck density now sits inside a general axiomatic theory of densities on $\sum 1/m_i$9. In the framework of Leonetti and Tringali, an upper density $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$00 satisfies normalization, monotonicity, subadditivity, and the translation-dilation law

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$01

and upper Buck density is explicitly listed among the standard examples (Leonetti et al., 2015). 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

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$02

for $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$03 and likewise for $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$04 (Leonetti et al., 2015). This strengthens mere surjectivity onto $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$05: 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 $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$06, finite decompositions

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$07

satisfying a common-refinement condition and a point-separation condition, and a finitely additive probability measure $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$08 on the algebra $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$09 generated by the decomposition atoms, one defines

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$10

A set is $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$11-measurable iff

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$12

and the restriction $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$13 is a finitely additive probability measure (Iacò et al., 2015). The classical Buck density is recovered by taking $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$14, $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$15 equal to the residue classes modulo $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$16, and $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$17. 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 (Iacò et al., 2015).

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

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$18

for every arithmetic quasi-density $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$19 (Leonetti et al., 2022). Moreover,

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$20

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

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$21

and uses it to prove that if $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$22 is non-empty and $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$23, then for each $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$24 there exists $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$25 such that, for every arithmetic quasi-density $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$26, both $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$27 and $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$28 are in the domain of $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$29 and

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$30

(Leonetti et al., 2022). This relies on periodic approximants modulo $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$31 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 $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$32 and

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$33

then there exist a positive integer

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$34

periodic sets $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$35, and exact finite-quotient formulas such as

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$36

together with a quasi-periodic or arithmetic-progression alternative in $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$37 (Hennecart, 2024). 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 $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$38, there exists $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$39 such that

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$40

and for $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$41 there exists $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$42 such that

$$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$43

so $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$44 has a Buck density while $$\mu^\ast(S)=\inf\Big\{\sum_{i=1}^k \frac{1}{m_i};\ S \subset \bigcup_{i=1}^k \bigl(r_i+(m_i)\bigr) \Big\},$$45 does not (Hennecart, 2024). These constructions reinforce a general point: Buck density is highly arithmetic, governed by congruence-class occupancy across scales rather than by interval growth alone.