--- title: Bohr Sets in Additive Combinatorics url: https://www.emergentmind.com/topics/bohr-sets type: topic --- # Bohr Sets in Additive Combinatorics Bohr sets are subsets of abelian groups defined by finitely many approximate character constraints; equivalently, they are inverse images of open subsets of finite-dimensional tori under group homomorphisms, and in the centered case they are neighborhoods of \(0\) for the Bohr topology [1512.01702][1608.01014]. They occupy a central position in additive combinatorics, ergodic Ramsey theory, harmonic analysis, Diophantine approximation, and the study of almost periodic and quasicrystalline structures, where they serve as canonical “structured” sets against which density, recurrence, and spectral phenomena are measured [2112.11997][2207.04150]. ## 1. Definitions, topology, and basic variants For a countable abelian group \(G\), a Bohr set may be presented as \[ B=\tau^{-1}(U), \] where \(\tau:G\to \mathbb{T}^n\) is a homomorphism with dense image and \(U\subset \mathbb{T}^n\) is open; if \(0_{\mathbb{T}^n}\in U\), then \(B\) is a **Bohr-zero set** [1512.01702]. In the equivalent character-theoretic formulation used throughout the literature, a Bohr set in a discrete abelian group is a finite intersection of sets of the form \[ \{g\in G: |\chi_j(g)-1|<\varepsilon\}, \] where \(\chi_j:G\to \mathcal S^1\) are characters [1608.01014]. For \(G=\mathbb{Z}\), this becomes \[ B(\alpha_1,\dots,\alpha_d;\varepsilon,\theta_1,\dots,\theta_d) =\Bigl\{n\in\mathbb{Z}:\|n\alpha_i-\theta_i\|_{\mathbb{R}/\mathbb{Z}}<\varepsilon \text{ for all }i\Bigr\}, \] and the centered case \(\theta_i=0\) is the standard Bohr neighborhood of \(0\) [2512.01997]. The **Bohr topology** on a discrete abelian group is the coarsest group topology making every character continuous; equivalently, it is induced by the embedding \[ G\to (\mathcal S^1)^{\widehat G},\qquad g\mapsto (\chi(g))_{\chi\in\widehat G}. \] Bohr neighborhoods are translates of Bohr sets, and Bohr sets form a neighborhood basis of \(0\) in this topology [1608.01014]. In compact abelian groups, the notation \[ B(\Lambda;\eta)=\{x\in G: |\chi(x)-1|<\eta \text{ for all }\chi\in\Lambda\} \] is standard; \(|\Lambda|\) is the **rank** and \(\eta\) the **radius** [2112.11997]. Two refinements recur in additive combinatorics. A **piecewise Bohr set** is an intersection \(B_0\cap T\), where \(B_0\) is Bohr and \(T\) is thick with upper Banach density \(1\) [1512.01702]. An **almost Bohr set** is a set of the form \(B\setminus E\), where \(B\) is Bohr and \(d^*(E)=0\) [2603.11376]. The latter notion is forced by Følner’s theorem on difference sets and is structurally weaker than genuine Bohr containment. The behavior of the Bohr topology depends strongly on the ambient group. In \(\mathbb F_p^\omega\), every nonzero element has order \(p\), every character takes values in \(p\)-th roots of unity, and Bohr-open sets are precisely unions of cosets of finite-index subgroups [1608.01014]. This torsion model sharply contrasts with the toral picture underlying \(\mathbb{Z}\), \(\mathbb{Z}^m\), and compact connected groups. ## 2. Structured largeness in sumsets and difference sets Bohr sets enter additive combinatorics as the structured output of density hypotheses. In \(\mathbb{Z}\), Bogolyubov’s theorem states that if \(A\subseteq\mathbb{Z}\) has positive upper Banach density, then \[ 2A-2A=A+A-A-A \] contains a Bohr set [2112.11997]. In compact abelian groups, the corresponding fourfold sum-difference \(A-A+A-A\) contains a Bohr set whenever \(\mu(A)>0\) [2112.11997]. Bergelson–Ruzsa’s three-term result replaces \(2A-2A\) by \(rA+sA+tA\) under the constraint \(r+s+t=0\), and the compact-group formulation extends this to commuting endomorphisms \(\phi_1,\phi_2,\phi_3\) with finite-index images and \[ \phi_1+\phi_2+\phi_3=0, \] yielding Bohr sets inside \(\phi_1(A)+\phi_2(A)+\phi_3(A)\) [2112.11997]. This compact-group theory has a countable discrete analogue. If \(G\) is a countable discrete abelian group and \(\phi_1,\phi_2,\phi_3:G\to G\) are commuting endomorphisms with finite-index images and \(\phi_1+\phi_2+\phi_3=0\), then for every \(A\subset G\) with \(d^*(A)>0\), the threefold sumset \[ \phi_1(A)+\phi_2(A)+\phi_3(A) \] contains a Bohr set whose rank and radius depend only on \(d^*(A)\) and the indices \([G:\phi_j(G)]\) [2207.04150]. Partition analogues also hold: for any finite partition \(G=\bigcup_{i=1}^r A_i\), some cell satisfies \[ \phi_1(A_i)+\phi_2(A_i)-\phi_2(A_i) \] contains a Bohr set, again with parameters depending only on \(r\) and the relevant indices [2207.04150]. A more localized phenomenon occurs in three-fold difference sets. If \(A\subseteq\mathbb{Z}\) has positive upper Banach density, then \(A+A-A\) contains Bohr neighborhoods of many elements of \(A\); more precisely, the radius and dimension depend only on \(d^*(A)\), and after removing a subset of \(A\) of arbitrarily small upper Banach density, every remaining \(a\in A\) satisfies that \[ A+A-A-a \] contains a Bohr-\((k,\eta)\) set [1608.02111]. The proof passes through a compact Kronecker factor, analyzes the convolution \(f*g*g_{-}\), and extracts a Bohr neighborhood from a short Fourier polynomial using a large-spectrum decomposition [1608.02111]. These results collectively establish a standard principle: sufficiently many additive combinations of a dense set force approximate periodicity visible on finitely many frequencies, and Bohr sets are the canonical container for that periodic structure. ## 3. Almost Bohr structure, DB and ABB sets, and torsion obstructions Følner’s theorem identifies the minimal Bohr content of difference sets: if \(A\subseteq G\) has \(d^*(A)>0\), then \(A-A\) contains an almost Bohr set \(B\setminus E\) with \(d^*(E)=0\) [2603.11376]. This leads to two expansion notions. A set \(S\subseteq G\) is **DB** if \(A-A+S\) contains a Bohr set for every positive-density \(A\), and **ABB** if \(A+S\) contains a Bohr set for every almost Bohr set \(A\) [2603.11376]. The ABB condition admits an exact characterization: \(S\) is ABB iff \(d^*(S\cap B)>0\) for every Bohr set \(B\), equivalently iff \(S\) meets every almost Bohr set [2603.11376]. In \(\mathbb{Z}\), several natural sparse sets are DB. The sets \[ \{n^2:n\in\mathbb N\},\qquad \{p-1:p\text{ prime}\},\qquad \{\lfloor n^c\rfloor:n\in\mathbb N\} \] with \(c>0\) all have the property that \(A-A+S\) contains a Bohr set for every \(A\) of positive upper Banach density; for intersective polynomial images and for polynomial values at primes of intersective polynomials of the second kind, the conclusion strengthens to the containment of a finite-index subgroup [2603.11376]. The torsion model \(G=\mathbb F_p^\omega\) shows that positive density alone does not force genuine Bohr neighborhoods in \(A-A\). There exists \(A\subseteq \mathbb F_p^\omega\) with positive upper Banach density such that \(A-A\) contains no Bohr neighborhood; for \(p=2\), one can arrange the stronger property that \(A-A\) contains no set of the form \(g+(B-B)\) with \(B\) piecewise syndetic [1608.01014]. The same paper constructs Bohr-dense sets \(S\) and positive-density sets \(A\) such that \(A+S\) is not piecewise Bohr, and for \(p=2\), not even piecewise syndetic [1608.01014]. These counterexamples show that Bohr structure is not a universal model for large additive behavior in torsion groups, even when density is high and Bohr-topological largeness is present. The contrast between the Følner almost-Bohr theorem and the \(\mathbb F_p^\omega\) counterexamples is structural: in some settings, adding a carefully chosen \(S\) upgrades almost Bohr structure to genuine Bohr structure, while in others even the raw difference set \(A-A\) can remain Bohr-thin. ## 4. Higher-order analogues: Nil\(_d\) Bohr sets, generalized polynomials, and recurrence Classical Bohr sets arise from rotations on compact abelian groups; higher-order analogues arise from nilsystems. A subset \(A\subseteq\mathbb Z\) is a **Nil\(_d\) Bohr\(_0\)-set** if there exist a minimal \(d\)-step nilsystem \((X,T)\), a point \(x_0\in X\), and an open neighborhood \(U\ni x_0\) such that \[ N(x_0,U)=\{n\in\mathbb Z:T^n x_0\in U\}\subset A \] [1407.1179]. For \(d=1\) this reduces to the usual Bohr\(_0\) notion. The paper “Nil Bohr-sets and almost automorphy of higher order” proves the exact arithmetic characterization \[ \mathcal F_{d,0}=\mathcal F\mathrm{GP}_d, \] identifying Nil\(_d\) Bohr\(_0\)-sets with families generated by generalized polynomial congruence conditions of degree at most \(d\) [1407.1179]. It also proves that for every Nil\(_d\) Bohr\(_0\)-set \(A\) there exists a syndetic set \(S\) such that \[ A\supset \{n\in\mathbb Z:S\cap(S-n)\cap\cdots\cap(S-dn)\neq\varnothing\}, \] thereby giving one half of a higher-order Bohr problem [1407.1179]. A complementary combinatorial comparison is provided by “Combinatorial properties of Nil-Bohr sets.” There, any Nil\(^d\) set is shown to be an SG\(^k\) set for all \(k\ge 4d\), so any Nil\(_d\)-Bohr\(_0\) set is necessarily \(\mathrm{SG}_k\) with \(k\) effectively bounded in terms of \(d\) [1507.07370]. The proof uses polynomial maps on the partial semigroup of finite subsets, Host–Kra cube groups, and an extraction theory for \(S_k\)-subsequences [1507.07370]. Bohr sets also define a recurrence notion independent of nilpotent structure. A subset \(S\subseteq\mathbb N\) is a set of \(d\)-dimensional Bohr recurrence if for every \(\alpha_1,\dots,\alpha_d\in\mathbb R\) and every \(\varepsilon>0\) there exists \(s\in S\) with \[ \|s\alpha_i\|_{\mathbb R/\mathbb Z}<\varepsilon\qquad (1\le i\le d). \] Recent work shows that every \(2\)-large set is a set of Bohr recurrence: if a set forces arbitrarily long monochromatic arithmetic progressions with common differences in every \(2\)-coloring, then it must intersect every finite-dimensional Bohr neighborhood [2512.01997]. This places Bohr recurrence as a linear topological-dynamical obstruction to combinatorial largeness. ## 5. Arithmetic, matrix, and finite-field manifestations In Diophantine approximation, Bohr sets encode simultaneous smallness of linear forms modulo \(1\). For irrational non-Liouville \(\alpha\), the inhomogeneous one-dimensional Bohr set \[ N_\gamma(\alpha,\rho)=\{n\in\mathbb N:n\rho N\), \(\min(N_1,N_2)>N^\varepsilon\), and \[ P(b;A_1,A_2;N_1,N_2)\subset N_\gamma(\alpha,\rho) \] [1703.07016]. This structure is the key combinatorial input in the inhomogeneous fibre version of Gallagher’s theorem. The higher-rank analogue replaces continued fractions by reduced successive minima. For \[ B_{\mathbf y}(N;\boldsymbol\delta)=\{n\in\mathbb Z: |n|\le N,\ |n\alpha_i-\gamma_i|<\delta_i\ (1\le i\le k-1)\}, \] the paper “Higher-rank Bohr sets and multiplicative diophantine approximation” develops an inner and outer generalized-arithmetic-progression theory of arbitrary rank. Under suitable lower bounds on the \(\delta_i\), \(B_{\mathbf y}(N;\boldsymbol\delta)\) contains a proper full-rank progression of size \(\gg \delta_1\cdots\delta_{k-1}N\), and also satisfies the cardinality bound \[ \#B_{\mathbf y}(N;\boldsymbol\delta)\ll \delta_1\cdots\delta_{k-1}N \] [1810.04558]. These estimates feed a Duffin–Schaeffer argument and yield higher-dimensional fibre refinements of Gallagher’s theorem, including inhomogeneous variants [1810.04558]. Bohr sets also interact with non-abelian algebraic actions. In the additive group \[ \Lambda=\operatorname{Mat}_d^0(\mathbb Z), \] any Bohr-zero non-periodic set intersects every \(SL_d(\mathbb Z)\)-conjugacy class: for every \(C\in\Lambda\) there exist \(A\in B\) and \(g\in SL_d(\mathbb Z)\) such that \[ C=g^{-1}Ag. \] Consequently, the characteristic polynomials realized by \(B\) coincide with those realized by all traceless integer matrices [1512.01702]. The proof uses an equidistribution theorem for an \(SL_d(\mathbb Z)\) random walk on the torus \[ \mathbb A_d=\operatorname{Mat}_d^0(\mathbb R)/\operatorname{Mat}_d^0(\mathbb Z), \] combined with spectral properties of piecewise Bohr sets [1512.01702]. In finite fields, additive Bohr sets also support analytic estimates. For \(B(T,\varepsilon)\subset\mathbb F_p\), one has \(|B+B|\le 4^d|B|\) when \(|T|=d\), and nontrivial multiplicative character sums satisfy analogues of the Pólya–Vinogradov and Burgess bounds [1409.7924]. This places finite-field Bohr sets at the intersection of additive structure and multiplicative cancellation. ## 6. Almost periodic sets, toral compactifications, and quasicrystals The term “Bohr” also governs a parallel theory of almost periodic discrete sets and measures. A Radon measure \(\mu\) on \(\mathbb R^n\) is **Bohr almost periodic** in Favorov’s sense if \(\mu*f\) is a Bohr almost periodic function for every \(f\in C_c(\mathbb R^n)\) [2107.10611]. When the Fourier transform \(\widehat\mu\) is supported on a finite-rank subgroup \(\Gamma\subset\mathbb R^n\), the measure is of **toral type**: there exists a toral compactification \[ \psi:\mathbb R^n\to \mathbb T^m \] with dense image, a compact set \(K=\overline{\psi(\Lambda)}\), and a measure \(\kappa\) on \(\mathbb T^m\) such that \[ \widehat\kappa=(\widehat\mu)\circ\widehat\psi \] [2107.10611]. For uniformly discrete Delone sets of toral type, each connected component of \(K\) is homeomorphic to \(\mathbb T^{m-n}\), embedded transversely to the foliation induced by \(\psi(\mathbb R^n)\), and the density of each component is given by \[ \operatorname{dens}(\Lambda_c)=|S_1/S|\;|\det(E^TM)| \] in terms of the compactification and the homotopy class of the embedding [2107.10611]. Earlier work showed that Bohr almost periodicity can be rigid in the presence of discrete difference structure: if a discrete set \(A\subset\mathbb R^p\) is Bohr almost periodic and of finite type, then \(A\) is an ideal crystal, \[ A=L+F, \] a finite union of translates of a full-rank lattice [1011.4036]. By contrast, a Besicovitch almost periodic Meyer set admits a weaker asymptotic lattice decomposition: there exists a full-rank lattice such that, for large balls, almost all points of \(A\) lie in equivalence classes with large intersection with those balls [1011.4036]. This distinction is central in quasicrystal theory, where strong Bohr almost periodicity can collapse a model to exact crystallinity. For regular Euclidean model sets, the Fourier side is especially explicit. If \(A=\Lambda(W)\) is a regular model set with Fourier module \(L^\circledast\), then the Fourier–Bohr coefficients \[ a(t)=\lim_{R\to\infty}\frac{1}{\mathrm{vol}(B_R)}\sum_{x\in A\cap B_R} e(-t\cdot x) \] exist for all \(t\), satisfy \[ a(t)=\mathrm{dens}(\mathcal L)\,\widehat{1_W}(-t^*)\qquad (t\in L^\circledast), \] and vanish for \(t\notin L^\circledast\) [2308.07105]. The significance of this result is methodological as well as structural: it yields the standard Fourier–Bohr formula for regular model sets by direct exponential-sum estimates and the Poisson summation formula, rather than by dynamical systems or harmonious-set theory [2308.07105]. Bohr sets therefore form a nexus connecting finitely many characters, toral compactifications, nilpotent higher-order recurrence, dense sumset structure, and the spectral analysis of model sets. Across these settings, the unifying principle is the same: finitely many frequency constraints create large sets with strong recurrence or approximate periodicity, and those sets control how algebraic, combinatorial, and spectral structure propagates. Source: https://www.emergentmind.com/topics/bohr-sets