Computable Følner Sequences & Monotilings

• Computable Følner sequences are algorithmically defined finite subsets in amenable groups that ensure effective amenability and support tiling and entropy computations.
• They extend classical Følner conditions by requiring explicit computability, enabling constructive proofs in group theory and operator algebras even when the word problem is unsolvable.
• This framework, including computable Følner monotilings and Tempelman sequences, provides actionable techniques for combinatorial tiling, ergodic theory, and spectral approximations in operator settings.

Searching arXiv for recent and foundational papers on computable Følner sequences, monotilings, and related constructive/operator-algebraic notions. Computable Følner sequences are effective versions of Følner sequences in amenable-group theory: instead of merely asserting the existence of finite subsets with asymptotically negligible boundary, one requires that these finite sets be generated algorithmically from the input data of the group. In the discrete-group setting, this notion is used to formalize effective amenability, to support algorithmic tilings and entropy computations, and to extend Brudno-type theorems beyond $\mathbb{Z}$ and $\mathbb{Z}^d$. Closely related but distinct notions appear in operator theory and $C^*$-algebras, where Følner sequences are sequences of finite-rank projections with asymptotically vanishing commutators. The literature therefore contains both a group-theoretic theory of computable Følner sets, sequences, and monotilings, and an operator-algebraic theory in which explicitness ranges from canonical constructions in concrete examples to nonconstructive existence theorems (Moriakov, 2015, Duda et al., 15 Sep 2025, Ara et al., 2013).

1. Definitions and formal variants

For a discrete group $\Gamma$, a Følner sequence is a sequence of finite subsets $(F_n)_{n\geq 1}$ such that for every finite $K \subset \Gamma$,

$\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$

as $n\to\infty$; equivalent boundary formulations are also standard, for example

$\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$

for every finite $R\subseteq \Gamma$. A group is amenable if it possesses a Følner sequence (Moriakov, 2015, Boretsky et al., 2019).

In computable-group settings, the basic effective strengthening is that the finite sets themselves are computable. For a computable group $\mathbb{Z}^d$0, a computable Følner sequence is a sequence $\mathbb{Z}^d$1 of finite subsets such that the characteristic function $\mathbb{Z}^d$2 is total computable; in the stronger “canonically computable” form used for monotilings, there is an algorithm which, on input $\mathbb{Z}^d$3, prints out the set $\mathbb{Z}^d$4 and halts. In the numbered-group framework of computably enumerable groups $\mathbb{Z}^d$5, an effective Følner sequence is a computable sequence of finite subsets of $\mathbb{Z}^d$6 coded via Gödel numbers, and computable amenability is phrased via an algorithm which, for each $\mathbb{Z}^d$7 with $\mathbb{Z}^d$8 and finite $\mathbb{Z}^d$9, produces a finite set $C^*$0 such that $C^*$1 is $C^*$2-Følner with respect to $C^*$3 and $C^*$4 (Moriakov, 2015, Duda et al., 15 Sep 2025).

A related but weaker notion is that of computable Følner sets. For a finitely generated group $C^*$5 with finite generating set $C^*$6, an $C^*$7-Følner set is a non-empty finite set $C^*$8 such that

$C^*$9

for all $\Gamma$0. The group has computable Følner sets with respect to $\Gamma$1 if there exists an algorithm which, on input $\Gamma$2, outputs a finite set of words whose image in $\Gamma$3 is $\Gamma$4-Følner. This definition is independent of the choice of generating set (Cavaleri, 2016).

The literature also distinguishes structurally significant subclasses of Følner sequences. A sequence $\Gamma$5 is tempered if there exists $\Gamma$6 such that

$\Gamma$7

for all $\Gamma$8, and Tempelman if there exists $\Gamma$9 such that

$(F_n)_{n\geq 1}$0

A Tempelman Følner sequence is always tempered, but not vice versa (Boretsky et al., 2019).

2. Effective amenability, computable Følner sets, and complexity

The modern theory of computable Følner sequences is closely tied to effective amenability. For computably enumerable groups, the finite-generation hypothesis can be removed from several basic equivalences: amenability, existence of computable Reiter functions, subrecursive Følner function, and $(F_n)_{n\geq 1}$1-amenability are equivalent. The same work also studies the arithmetic complexity of families of effective Følner sequences and discusses extensions to metric groups (Duda et al., 15 Sep 2025).

In finitely generated groups, computable Følner sets give an algorithmic version of amenability that does not presuppose solvability of the word problem. If a group is amenable and has solvable word problem, then Følner sets are computable by enumerating finite subsets and verifying the Følner condition. The notable counterpoint is that computable Følner sets can exist even when the word problem is unsolvable: the Kharlampovich groups $(F_n)_{n\geq 1}$2, finitely presented solvable groups with unsolvable word problem, admit computable Følner sets by an explicit construction (Cavaleri, 2016).

Cavaleri’s construction for Kharlampovich groups is built from explicit product sets. If $(F_n)_{n\geq 1}$3 is a finite subset generating a factor $(F_n)_{n\geq 1}$4, then

$(F_n)_{n\geq 1}$5

is shown to be $(F_n)_{n\geq 1}$6-Følner in $(F_n)_{n\geq 1}$7, and the group-level Følner sets are given by

$(F_n)_{n\geq 1}$8

The same paper proves preservation results for extensions and gives upper bounds for Følner functions in those contexts (Cavaleri, 2016).

The 2025 work further shows that the set of all effective Følner sequences of a computable group is a $(F_n)_{n\geq 1}$9 subset in the arithmetic hierarchy, and is $K \subset \Gamma$0-complete for some abelian groups, including $K \subset \Gamma$1. It also proves that for any total computable function $K \subset \Gamma$2, there exists a computable $K \subset \Gamma$3 such that the modulus of convergence of the means

$K \subset \Gamma$4

is not bounded by $K \subset \Gamma$5, indeed not by any primitive recursive function. This places a sharp limit on how much uniform quantitative control can be expected from computable Følner data alone (Duda et al., 15 Sep 2025).

Concrete examples remain central. In $K \subset \Gamma$6 under any one-to-one computable enumeration,

$K \subset \Gamma$7

forms an effective Følner sequence, with each term a $K \subset \Gamma$8-Følner set (Duda et al., 15 Sep 2025).

3. Computable Følner monotilings and strengthened effective structure

A computable Følner monotiling is a substantial strengthening of a computable Følner sequence. A monotiling in $K \subset \Gamma$9 is a pair $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$0 where $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$1 is a finite set and $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$2 is such that the family $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$3 is a partition of $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$4 into disjoint left-translates of $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$5. A Følner monotiling is a sequence $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$6 such that $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$7 is a Følner sequence, and it is computable if $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$8 is canonically computable and $\frac{|F_nK\setminus F_n|}{|F_n|}\to 0$9 is computable, meaning that the membership function $n\to\infty$0 is computable in both variables (Moriakov, 2015, Moriakov, 2015).

The normality conditions introduced in the first Brudno paper are

• $n\to\infty$1,
• $n\to\infty$2 for all $n\to\infty$3.

Every computable Følner monotiling can be normalized to a computable normal Følner monotiling. The second Brudno paper refines this further to computable regular symmetric Følner monotilings: the sequence must be tempered and two-sided, each $n\to\infty$4 must be a good weight for pointwise ergodic averages along $n\to\infty$5, $n\to\infty$6, $n\to\infty$7 for every $n\to\infty$8, and $n\to\infty$9 for all $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$0 (Moriakov, 2015, Moriakov, 2015).

This framework provides effective tilings in concrete groups. For $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$1 one has

$\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$2

and $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$3 partitions $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$4. For groups of unipotent upper-triangular matrices $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$5, the papers state that particularly nice computable regular symmetric Følner monotilings exist and that the required computing algorithms can be provided explicitly. In $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$6, for instance, $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$7 consists of matrices with $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$8 divisible by $\lim_n \frac{|\partial_R F_n|}{|F_n|}=0$9 and $R\subseteq \Gamma$0 divisible by $R\subseteq \Gamma$1, while $R\subseteq \Gamma$2 is given by the ranges $R\subseteq \Gamma$3 and $R\subseteq \Gamma$4 (Moriakov, 2015, Moriakov, 2015).

The class of groups admitting computable Følner monotilings is stable under computable extensions. If

$R\subseteq \Gamma$5

is an exact sequence of computable groups with computable homomorphisms, and if $R\subseteq \Gamma$6 and $R\subseteq \Gamma$7 admit computable normal Følner monotilings, then so does $R\subseteq \Gamma$8. The construction lifts tiles from the quotient, selects computable sections, and fills fibers with the monotiling in the kernel (Moriakov, 2015).

A further constructive refinement is available at the level of subsequences: given a canonically computable Følner sequence, an algorithm computably selects a tempered subsequence (Moriakov, 2015).

4. Entropy, Kolmogorov complexity, and symbolic dynamics

Computable Følner monotilings were introduced to make Brudno-type entropy theorems available for actions of groups beyond the classical one-dimensional setting. In the topological version, if $R\subseteq \Gamma$9 is a computable group with a computable normal Følner monotiling $\mathbb{Z}^d$00, if $\mathbb{Z}^d$01 is a subshift of $\mathbb{Z}^d$02, and if $\mathbb{Z}^d$03 is the associated word presheaf, then

$\mathbb{Z}^d$04

where

$\mathbb{Z}^d$05

is the asymptotic Kolmogorov complexity of sections. Entropy is computed by

$\mathbb{Z}^d$06

The monotiling structure is essential because it allows effective decomposition and reconstruction of words over large Følner sets (Moriakov, 2015).

The measure-theoretic extension requires the more restrictive regular symmetric monotiling. For a subshift $\mathbb{Z}^d$07, with $\mathbb{Z}^d$08 computable and admitting a computable regular symmetric Følner monotiling, and for any ergodic invariant measure $\mathbb{Z}^d$09,

$\mathbb{Z}^d$10

Here $\mathbb{Z}^d$11 is the asymptotic Kolmogorov complexity of the section $\mathbb{Z}^d$12 along the computable Følner sequence $\mathbb{Z}^d$13 from the monotiling (Moriakov, 2015).

These theorems clarify why computable Følner sequences alone are often insufficient for algorithmic symbolic dynamics. The additional data of computable centers $\mathbb{Z}^d$14 turns large finite sets into usable coding domains. The papers explicitly state that a computable Følner monotiling is a strengthening of a computable Følner sequence: it not only gives effective Følner sets, but also a computable way to partition the whole group by translates of these sets (Moriakov, 2015).

A plausible implication is that computability requirements in entropy theory are not concentrated in amenability itself, but in the availability of effective combinatorial decompositions adapted to the Følner geometry.

5. Constructive tilings along Tempelman Følner sequences

A different constructive direction is developed for increasing Tempelman Følner sequences. For pmp actions of amenable groups, the paper proves that the tiling property holds along increasing Tempelman Følner sequences, and that this property directly implies the pointwise ergodic theorem. The averaging operators are

$\mathbb{Z}^d$15

and the theorem states that if a group has the tiling property along $\mathbb{Z}^d$16, then for any pmp action on $\mathbb{Z}^d$17, ergodicity is equivalent to almost-everywhere convergence of these averages to $\mathbb{Z}^d$18 for every $\mathbb{Z}^d$19 (Boretsky et al., 2019).

The technical core is a multi-scale combinatorial tiling argument based on an iterated Vitali covering lemma. If $\mathbb{Z}^d$20 and $\mathbb{Z}^d$21 is finite, one can select a disjoint family $\mathbb{Z}^d$22 whose union $\mathbb{Z}^d$23 satisfies

$\mathbb{Z}^d$24

where $\mathbb{Z}^d$25 is the Tempelman constant. Iterating over scales covers a constant fraction of what remains at each stage; after $\mathbb{Z}^d$26 scales, the uncovered fraction is at most

$\mathbb{Z}^d$27

This yields arbitrarily efficient tilings (Boretsky et al., 2019).

The paper explicitly emphasizes the elementary and algorithmic character of the construction: it gives a step-by-step combinatorial process, uses only the Tempelman condition and the group operation, and avoids heavy analysis. It further states that, because the construction is recursive in the data $\mathbb{Z}^d$28 group operation, and $\mathbb{Z}^d$29, Tempelman Følner sequences permit computable tiling procedures. This does not amount to a general theory of computable Følner sequences for all amenable groups, but it identifies a class of Følner sequences for which constructive ergodic-theoretic arguments are available (Boretsky et al., 2019).

A common misconception is that ergodic theorems along Følner sequences are intrinsically analytic. In the Tempelman setting, the cited work shows instead that the argument can be fundamentally combinatorial and constructive.

6. Operator-theoretic and $\mathbb{Z}^d$30-algebraic analogues

In operator theory and operator algebras, the term Følner sequence refers to finite-rank projections rather than finite subsets of a group. For a set of operators $\mathbb{Z}^d$31, a Følner sequence is a sequence $\mathbb{Z}^d$32 of nonzero finite-rank orthogonal projections such that

$\mathbb{Z}^d$33

where $\mathbb{Z}^d$34 is the Hilbert–Schmidt norm. A proper Følner sequence is increasing and converges strongly to the identity (Ara et al., 2013, Ara et al., 2012).

This notion supports trace and spectral approximation. If $\mathbb{Z}^d$35 is a unital separable $\mathbb{Z}^d$36-algebra with amenable trace $\mathbb{Z}^d$37 and $\mathbb{Z}^d$38, there exists a proper Følner sequence $\mathbb{Z}^d$39 such that

$\mathbb{Z}^d$40

The same papers characterize Følner $\mathbb{Z}^d$41-algebras by sequences of u.c.p. maps

$\mathbb{Z}^d$42

satisfying

$\mathbb{Z}^d$43

However, the general existence proof is abstract rather than algorithmic; the authors explicitly remark that it “gives in general no clue of what the matrix approximations of concrete operators are” (Ara et al., 2012).

At the same time, several canonical explicit constructions are available. For the unilateral shift $\mathbb{Z}^d$44 on $\mathbb{Z}^d$45, the projections onto $\mathbb{Z}^d$46 form a proper Følner sequence, and

$\mathbb{Z}^d$47

This construction is fully explicit and algorithmic. More generally, every essentially normal operator has a proper Følner sequence, with the proof based on Brown–Douglas–Fillmore theory and constructive combination rules such as the absorbing property for direct sums (Ara et al., 2013).

Crossed products yield another explicit family. If $\mathbb{Z}^d$48 has a Følner sequence $\mathbb{Z}^d$49, if $\mathbb{Z}^d$50 is a countable discrete amenable group with Følner sets $\mathbb{Z}^d$51, and if the action $\mathbb{Z}^d$52 satisfies the compatibility condition

$\mathbb{Z}^d$53

then the projections

$\mathbb{Z}^d$54

form a canonical Følner sequence for the crossed product $\mathbb{Z}^d$55. The paper applies this to the rotation algebra and to the $\mathbb{Z}^d$56-algebra of bounded Jacobi operators, where the resulting finite sections support spectral approximation (Lledó, 2010).

The operator-algebraic literature therefore parallels the group-theoretic one in a characteristic way: explicit and computable Følner constructions exist in structured examples, but general existence theorems need not furnish effective algorithms.