Buck Measurable Sets: Theory & Applications

• Buck measurable sets are subsets of ℕ or ℤ whose outer and inner Buck densities coincide using finite unions of arithmetic progressions.
• They extend natural density by employing residue-class formulations and factorial moduli, with practical implications in small-set theory and additive combinatorics.
• This framework enables explicit constructions, controlled additivity, and links to uniform distribution, offering a robust tool in arithmetic density studies.

Buck measurable sets are subsets of $\mathbb N$ or $\mathbb Z$ whose outer and inner Buck densities, defined through approximation by finite unions of arithmetic progressions, coincide. In Buck’s measure-theoretic approach to density, unions of residue classes serve as the basic measurable objects, each such union having exact density $|R|/m$. This framework extends natural density on periodic sets, admits several equivalent residue-covering formulations, and has been developed in directions including small-set theory, additive combinatorics, generalized decomposition systems, and uniform distribution (Leonetti et al., 2019, Leonetti et al., 2022, Iacò et al., 2015).

1. Definition and basic framework

For $H\in\{\mathbb N,\mathbb Z\}$, let $A_H$ denote the family of all finite unions of arithmetic progressions of $H$. A standard notation is

$$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$

where $m\in\mathbb N^+$ and $R\subseteq\{0,1,\dots,m-1\}$. The upper Buck density of $X\subseteq H$ is

$\mathbb Z$0

and the lower Buck density is

$\mathbb Z$1

A set $\mathbb Z$2 is Buck measurable if $\mathbb Z$3; the common value is denoted $\mathbb Z$4. On the algebra $\mathbb Z$5, Buck measure agrees with the density of the corresponding union of residue classes: if $\mathbb Z$6, then $\mathbb Z$7 (Leonetti et al., 2019).

Equivalent notation appears in later work. One common variant defines

$\mathbb Z$8

where $\mathbb Z$9 is the family of finite unions of arithmetic progressions and $|R|/m$0 is upper asymptotic density. Another uses factorial moduli and covering numbers $|R|/m$1, the minimal number of residue classes modulo $|R|/m$2 covering $|R|/m$3, with upper and lower Buck densities given by $|R|/m$4 and $|R|/m$5, respectively (Leonetti et al., 2022).

In the original setting, Buck’s construction is an outer-density procedure built from arithmetic progressions. The central object is not $|R|/m$6-additivity on the full power set, but a congruence-based outer/inner approximation scheme. This arithmetic character is the source of both the strength and the limitations of Buck measurability (Leonetti et al., 2019).

2. Residue-class characterizations

A key feature of Buck density is that it can be expressed directly in terms of the residue classes met or fully contained by a set. For $|R|/m$7, define

$|R|/m$8

and

$|R|/m$9

Then

$H\in\{\mathbb N,\mathbb Z\}$0

Thus the upper Buck density records how small a proportion of residue classes modulo some modulus can cover $H\in\{\mathbb N,\mathbb Z\}$1, while the lower Buck density records how large a proportion of complete residue classes can be embedded in $H\in\{\mathbb N,\mathbb Z\}$2 (Leonetti et al., 2019).

A related formulation, used in work on upper Buck density in $H\in\{\mathbb N,\mathbb Z\}$3, considers the image $H\in\{\mathbb N,\mathbb Z\}$4 of $H\in\{\mathbb N,\mathbb Z\}$5 modulo $H\in\{\mathbb N,\mathbb Z\}$6, together with

$H\in\{\mathbb N,\mathbb Z\}$7

For a multiplicatively increasing and exhaustive sequence $H\in\{\mathbb N,\mathbb Z\}$8, one has

$H\in\{\mathbb N,\mathbb Z\}$9

This shows that upper Buck density can be recovered from the asymptotic fraction of residue classes attained modulo sufficiently divisible moduli (Hennecart, 2024).

The factorial-covering perspective is equivalent in the regimes relevant to Buck measurability. In particular, $A_H$0 if and only if $A_H$1, meaning that $A_H$2 occupies a vanishing fraction of residue classes modulo $A_H$3 (Leonetti et al., 2022). A further reformulation uses “reminder systems”: if $A_H$4 is a sequence such that every fixed divisor eventually divides $A_H$5, then

$A_H$6

where $A_H$7 counts the number of residues modulo $A_H$8 represented in $A_H$9 (Pasteka, 1 Aug 2025).

3. Relation to natural density and measure-theoretic structure

Buck density is designed to dominate upper natural density and to be dominated below by lower natural density. For every $H$0,

$H$1

Hence Buck outer density is at least as large as upper asymptotic density, and Buck inner density is at most lower asymptotic density (Pasteka, 1 Aug 2025).

A classical implication, emphasized repeatedly in the modern literature, is that if asymptotic density exists, then Buck measurability follows and the two densities agree. One later measure-theoretic formulation of Niven’s theorem states the converse as well: $H$2 is Buck measurable if and only if its asymptotic density exists, and then

$H$3

This formulation presents Buck measurability as equivalent to existence of natural density in that setting (Pasteka, 1 Aug 2025). At minimum, the cited sources agree that Buck density extends asymptotic density on all sets where the latter exists, and exactly on every finite union of residue classes (Leonetti et al., 2022).

Buck measure is finitely additive on the algebra of measurable sets. There is also a controlled countable-additivity statement: if $H$4 are disjoint Buck measurable sets and

$H$5

then $H$6 is Buck measurable and

$H$7

A more general majorant condition via residue-coverage bounds yields the same conclusion (Pasteka, 1 Aug 2025).

The topological interpretation is equally important. The identity

$H$8

identifies Buck outer density with the Haar probability of the closure of $H$9 in the compact ring of polyadic integers. In this formulation, Buck measurability corresponds to Carathéodory measurability, while the criterion

$$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$0

serves as a convenient equivalent condition (Pasteka, 1 Aug 2025).

4. Buck-null sets, small sets, and instructive examples

A major structural result identifies Buck-null sets with sets that vanish under every upper quasi-density. An upper quasi-density $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$1 on $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$2 is a normalized, subadditive, dilation- and translation-invariant set function; if it is also monotone, it is an upper density. The central theorem states that for all $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$3,

$$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$4

for every upper quasi-density $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$5, and therefore $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$6 is small, meaning $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$7 for every upper quasi-density, if and only if $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$8 (Leonetti et al., 2019).

The class of small, equivalently Buck-null, sets is an ideal: it is closed under subsets and finite unions, contains all finite sets, is invariant under dilations and translations, and is independent of whether one works in $$C(m,R):=\bigcup_{r\in R}\{n\in H:n\equiv r\pmod m\},$$9 or $m\in\mathbb N^+$0. At the same time, it is not closed under products or sums. The cited exposition notes, for example, that there exist small sets $m\in\mathbb N^+$1 such that $m\in\mathbb N^+$2 or $m\in\mathbb N^+$3 is not small; one explicit phenomenon is that $m\in\mathbb N^+$4 is small, but $m\in\mathbb N^+$5 has Buck measure $m\in\mathbb N^+$6 (Leonetti et al., 2019).

This framework yields a large class of concrete Buck-null sets. If

$m\in\mathbb N^+$7

where $m\in\mathbb N^+$8 counts prime factors with multiplicity, then $m\in\mathbb N^+$9, and hence $R\subseteq\{0,1,\dots,m-1\}$0, is small. If $R\subseteq\{0,1,\dots,m-1\}$1 is non-linear, then $R\subseteq\{0,1,\dots,m-1\}$2 is small. If

$R\subseteq\{0,1,\dots,m-1\}$3

and $R\subseteq\{0,1,\dots,m-1\}$4 is not a perfect square or $R\subseteq\{0,1,\dots,m-1\}$5, then $R\subseteq\{0,1,\dots,m-1\}$6 is small. Digit-avoidance sets in a fixed base are likewise Buck-null (Leonetti et al., 2019).

A standard cautionary example shows that Buck-nullity is strictly stronger than zero asymptotic density. The set

$R\subseteq\{0,1,\dots,m-1\}$7

has $R\subseteq\{0,1,\dots,m-1\}$8 because $R\subseteq\{0,1,\dots,m-1\}$9 for every $X\subseteq H$0, even though $X\subseteq H$1 has zero asymptotic density. The point is that $X\subseteq H$2 meets every residue class modulo every modulus, so it is arithmetically large in Buck’s sense despite being sparse in the classical counting sense (Leonetti et al., 2019).

5. Explicit constructions of Buck measurable sets

Beyond null sets, the theory provides direct constructions of Buck measurable sets with prescribed measure. A general device begins with an increasing sequence $X\subseteq H$3 satisfying $X\subseteq H$4, and Buck measurable sets $X\subseteq H$5 such that every element of $X\subseteq H$6 is relatively prime to all $X\subseteq H$7. Then

$X\subseteq H$8

is Buck measurable and

$X\subseteq H$9

The divisibility chain makes the residue-counting systems compatible, while the coprimality condition controls overlaps among congruence classes (Pasteka, 1 Aug 2025).

A concrete realization of arbitrary measure uses the dyadic expansion $\mathbb Z$00 with strictly increasing $\mathbb Z$01. Let $\mathbb Z$02 be the odd integers, so $\mathbb Z$03, and define

$\mathbb Z$04

Then $\mathbb Z$05 is Buck measurable and $\mathbb Z$06. This reproduces the classical fact that Buck’s framework realizes every value in $\mathbb Z$07 as the measure of some measurable set (Pasteka, 1 Aug 2025).

Prime-adic prescriptions furnish another class of examples. For a prime $\mathbb Z$08 and an increasing set $\mathbb Z$09, let

$\mathbb Z$10

Then

$\mathbb Z$11

and

$\mathbb Z$12

For finitely many distinct primes $\mathbb Z$13 with exponent sets $\mathbb Z$14, the corresponding simultaneous prescription set is Buck measurable with

$\mathbb Z$15

These formulas exhibit Buck measure as highly compatible with multiplicative arithmetic structure (Pasteka, 1 Aug 2025).

The same methods yield zero-measure constructions via a Buck-form of Niven’s criterion. If $\mathbb Z$16 is a sequence of primes with $\mathbb Z$17, and

$\mathbb Z$18

then

$\mathbb Z$19

Applications include sets of integers with at most $\mathbb Z$20 prescribed odd prime exponents, sets with at most $\mathbb Z$21 prime factors, and the set

$\mathbb Z$22

all of which have Buck measure $\mathbb Z$23 in the stated settings (Pasteka, 1 Aug 2025).

6. Additive, comparative, and generalized developments

Buck measurability has become a useful intermediary among arithmetic densities. For an arbitrary arithmetic quasi-density $\mathbb Z$24, one has

$\mathbb Z$25

and $\mathbb Z$26 for every $\mathbb Z$27. This means that once a set is known to be Buck measurable, its value is automatically fixed across the entire class of arithmetic quasi-densities considered in that framework (Leonetti et al., 2022).

This transfer principle underlies a sumset theorem of Leonetti and Tringali. If $\mathbb Z$28 is nonempty and $\mathbb Z$29, equivalently $\mathbb Z$30, then for every $\mathbb Z$31 there exists $\mathbb Z$32 such that

$\mathbb Z$33

for every arithmetic quasi-density $\mathbb Z$34, with both $\mathbb Z$35 and $\mathbb Z$36 belonging to $\mathbb Z$37. Primes and perfect powers are listed as examples of such $\mathbb Z$38 (Leonetti et al., 2022).

Upper Buck density also interacts with additive-combinatorial structure through periodicity. In the integers, periodic subsets of $\mathbb Z$39 are exactly finite unions of arithmetic progressions, and recent work establishes a Kneser-type theorem for upper Buck density, comparing it with corresponding results for upper Banach density and constructing sequences with counterintuitive behavior for the Buck densities of $\mathbb Z$40 and $\mathbb Z$41 (Hennecart, 2024).

A broader generalization extends Buck’s measure density from finite unions of arithmetic progressions to arbitrary subsets of $\mathbb Z$42 defined by a prescribed system of decompositions. This enlargement produces new examples and links the theory to uniform distribution, showing that Buck-type measurability is not confined to the classical residue-class algebra but can be transported to more general decomposition schemes (Iacò et al., 2015).

Taken together, these developments position Buck measurable sets at the intersection of density theory, congruence methods, additive number theory, and uniform distribution. Their defining feature is not mere largeness or smallness in counting terms, but arithmetic regularity under approximation by structured residue systems.