--- title: 'Buck Measurable Sets: Theory & Applications' url: https://www.emergentmind.com/topics/buck-measurable-sets type: topic --- # Buck Measurable Sets: Theory & Applications 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 [1905.08075] [2212.11176] [1502.01611]. ## 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 \[ \mu_B^*(X):=\inf\Big\{\frac{|R|}{m}:X\subseteq C(m,R)\Big\}, \] and the lower Buck density is \[ \mu_{B,*}(X):=\sup\Big\{\frac{|R|}{m}:C(m,R)\subseteq X\Big\}. \] A set \(X\) is Buck measurable if \(\mu_B^*(X)=\mu_{B,*}(X)\); the common value is denoted \(\mu_B(X)\). On the algebra \(A_H\), Buck measure agrees with the density of the corresponding union of residue classes: if \(A=C(m,R)\), then \(\mu_B(A)=|R|/m=d_H(A)\) [1905.08075]. Equivalent notation appears in later work. One common variant defines \[ b^\star(X):=\inf\{\overline d(A):A\in\mathcal A,\ X\subseteq A\}, \] where \(\mathcal A\) is the family of finite unions of arithmetic progressions and \(\overline d\) is upper asymptotic density. Another uses factorial moduli and covering numbers \(\beta_n(S)\), the minimal number of residue classes modulo \(n!\) covering \(S\), with upper and lower Buck densities given by \(\limsup \beta_n(S)/n!\) and \(\liminf \beta_n(S)/n!\), respectively [2212.11176]. In the original setting, Buck’s construction is an outer-density procedure built from arithmetic progressions. The central object is not \(\sigma\)-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 [1905.08075]. ## 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 \(X\subseteq H\), define \[ r_m(X):=\big|\{r\in\{0,1,\dots,m-1\}:X\cap(m\cdot H+r)\neq\varnothing\}\big|, \] and \[ s_m(X):=\big|\{r\in\{0,1,\dots,m-1\}:m\cdot H+r\subseteq X\}\big|. \] Then \[ \mu_B^*(X)=\inf_{m\ge 1}\frac{r_m(X)}{m}, \qquad \mu_{B,*}(X)=\sup_{m\ge 1}\frac{s_m(X)}{m}. \] Thus the upper Buck density records how small a proportion of residue classes modulo some modulus can cover \(X\), while the lower Buck density records how large a proportion of complete residue classes can be embedded in \(X\) [1905.08075]. A related formulation, used in work on upper Buck density in \(\mathbb N\), considers the image \(X^{(m)}\subseteq\{0,1,\dots,m-1\}\) of \(X\) modulo \(m\), together with \[ X^\infty_{(m)}=\{k:(k+m\mathbb N)\cap X\ \text{is infinite}\}, \qquad X^*_{(m)}=\{k:(k+m\mathbb N)\setminus X\ \text{is finite}\}. \] For a multiplicatively increasing and exhaustive sequence \((m_n)\), one has \[ \overline b(X)=\lim_{n\to\infty}\frac{|X^{(m_n)}|}{m_n} = \lim_{n\to\infty}\frac{|X^\infty_{(m_n)}|}{m_n}. \] This shows that upper Buck density can be recovered from the asymptotic fraction of residue classes attained modulo sufficiently divisible moduli [2410.13275]. The factorial-covering perspective is equivalent in the regimes relevant to Buck measurability. In particular, \(b^\star(X)=0\) if and only if \(\beta_n(X)=o(n!)\), meaning that \(X\) occupies a vanishing fraction of residue classes modulo \(n!\) [2212.11176]. A further reformulation uses “reminder systems”: if \((B_N)\) is a sequence such that every fixed divisor eventually divides \(B_N\), then \[ \mu^\ast(S)=\lim_{N\to\infty}\frac{R(S:B_N)}{B_N}, \] where \(R(S:m)\) counts the number of residues modulo \(m\) represented in \(S\) [2508.00538]. ## 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 \(A\subseteq\mathbb N\), \[ \mu_*(A)\le \underline d(A)\le \overline d(A)\le \mu^*(A). \] Hence Buck outer density is at least as large as upper asymptotic density, and Buck inner density is at most lower asymptotic density [2508.00538]. 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: \(A\subseteq\mathbb N\) is Buck measurable if and only if its asymptotic density exists, and then \[ \mu(A)=d(A). \] This formulation presents Buck measurability as equivalent to existence of natural density in that setting [2508.00538]. 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 [2212.11176]. Buck measure is finitely additive on the algebra of measurable sets. There is also a controlled countable-additivity statement: if \(A_1,A_2,\dots\) are disjoint Buck measurable sets and \[ \lim_{N\to\infty}\mu^\ast\Big(\bigcup_{k=N}^\infty A_k\Big)=0, \] then \(\bigcup_{k=1}^\infty A_k\) is Buck measurable and \[ \mu\Big(\bigcup_{k=1}^\infty A_k\Big)=\sum_{k=1}^\infty \mu(A_k). \] A more general majorant condition via residue-coverage bounds yields the same conclusion [2508.00538]. The topological interpretation is equally important. The identity \[ \mu^\ast(S)=P(\overline S) \] identifies Buck outer density with the Haar probability of the closure of \(S\) in the compact ring of polyadic integers. In this formulation, Buck measurability corresponds to Carathéodory measurability, while the criterion \[ \mu^\ast(A)+\mu^\ast(\mathbb N\setminus A)=1 \] serves as a convenient equivalent condition [2508.00538]. ## 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 \(\mu^\ast\) on \(H\) 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 \(X\subseteq H\), \[ 0\le \mu^\ast(X)\le \mu_B^\ast(X) \] for every upper quasi-density \(\mu^\ast\), and therefore \(X\) is small, meaning \(\mu^\ast(X)=0\) for every upper quasi-density, if and only if \(\mu_B^\ast(X)=0\) [1905.08075]. 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 \(\mathbb N\) or \(\mathbb Z\). At the same time, it is not closed under products or sums. The cited exposition notes, for example, that there exist small sets \(X,Y\) such that \(XY\) or \(X+X\) is not small; one explicit phenomenon is that \(Q=\{x^2+y^2:x,y\in H\}\) is small, but \(Q+Q=\mathbb N\) has Buck measure \(1\) [1905.08075]. This framework yields a large class of concrete Buck-null sets. If \[ X^{(k)}=\{n\in H:\Omega(n)=k\}, \qquad Y^{(k)}=\{n\in H:\Omega(n)\le k\}, \] where \(\Omega\) counts prime factors with multiplicity, then \(Y^{(k)}\), and hence \(X^{(k)}\), is small. If \(F\in\mathbb Z[x]\) is non-linear, then \(F(H)\) is small. If \[ X=\{ax^2+bxy+cy^2:x,y\in H\}, \qquad D=b^2-4ac, \] and \(D\) is not a perfect square or \(D=0\), then \(X\) is small. Digit-avoidance sets in a fixed base are likewise Buck-null [1905.08075]. A standard cautionary example shows that Buck-nullity is strictly stronger than zero asymptotic density. The set \[ X=\{h!+h:h\in\mathbb N\} \] has \(\mu_B^\ast(X)=1\) because \(r_k(X)=k\) for every \(k\), even though \(X\) has zero asymptotic density. The point is that \(X\) meets every residue class modulo every modulus, so it is arithmetically large in Buck’s sense despite being sparse in the classical counting sense [1905.08075]. ## 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 \((b_i)\) satisfying \(b_i\mid b_{i+1}\), and Buck measurable sets \(H_i\) such that every element of \(\bigcup H_i\) is relatively prime to all \(b_i\). Then \[ H=\bigcup_{i=1}^\infty b_iH_i \] is Buck measurable and \[ \mu(H)=\sum_{i=1}^\infty \frac{\mu(H_i)}{b_i}. \] The divisibility chain makes the residue-counting systems compatible, while the coprimality condition controls overlaps among congruence classes [2508.00538]. A concrete realization of arbitrary measure uses the dyadic expansion \(\alpha=\sum_k2^{-n_k}\) with strictly increasing \(n_k\). Let \(\mathbf O\) be the odd integers, so \(\mu(\mathbf O)=\tfrac12\), and define \[ B_\alpha=\bigcup_k 2^{\,n_k-1}\mathbf O. \] Then \(B_\alpha\) is Buck measurable and \(\mu(B_\alpha)=\alpha\). This reproduces the classical fact that Buck’s framework realizes every value in \([0,1]\) as the measure of some measurable set [2508.00538]. Prime-adic prescriptions furnish another class of examples. For a prime \(p\) and an increasing set \(E=\{e_1