June 04, 2026
The goal of this short note is to prove a unified abstract regularity lemma which recovers Szemerédi’s graph regularity lemma, Green’s arithmetic regularity lemma, and a regularity lemma for Boolean functions as direct corollaries.
Regularity lemmas are among the basic structural tools of modern discrete mathematics. They assert, in different settings, that an arbitrary object can be decomposed into a bounded-complexity structured part and a pseudorandom remainder. The paradigmatic example is Szemerédi’s graph regularity lemma [1]; closely related regularity statements appear in additive combinatorics, most notably Green’s arithmetic regularity lemma [2], and in the analysis of Boolean functions, for instance in the regularity lemma used by Minzer [3] to recover the result of Dinur and Friedgut asserting that an intersecting family lies in a junta [4]. These regularity lemmas have found a remarkably broad range of applications across discrete mathematics, playing a central role in developments in graph theory, additive combinatorics, theoretical computer science, and the analysis of Boolean functions.
A useful distinction is between weak and strong regularity lemmas. Weak regularity lemmas, such as the Frieze–Kannan lemma, admit a rather general abstract formulation; one such formulation is due to Trevisan, Tulsiani and Vadhan [5]. They typically yield quantitatively much stronger bounds at the expense of providing a correspondingly weaker structural description. For strong regularity lemmas, the situation is less uniform. There are abstract approaches, including the probabilistic framework of Tao [6] and the semiring framework of Bollobás and Nikiforov [7], but these do not give, in a completely direct way, all of the standard examples one would like to recover, especially the Fourier-analytic form of Green’s arithmetic regularity lemma.
The purpose of this short note is to isolate an elementary abstract statement from which the most standard strong regularity lemmas follow with essentially no additional work. This also clarifies why the same energy-increment proof is operating in graph, Fourier-analytic, and Boolean-function settings. For completeness, in an appendix, we also revisit the frameworks of Tao and of Bollobás–Nikiforov, showing how they can be recovered from our abstract formulation.
Organization. The remainder of this note is organized as follows. In Section 2, we state and prove our unified abstract regularity lemma (Theorem 1). In Section 3, we derive Green’s arithmetic regularity lemma from Theorem 1. In Section 4, we derive Szemerédi’s graph regularity lemma from the same abstract theorem. In Section 5, we obtain the
aforementioned regularity lemma for Boolean functions. In the appendix, we show how our framework recovers the abstract regularity lemmas of Bollobás and Nikiforov, and of Tao.
Acknowledgments. We are grateful to our supervisors, Imre Leader and Timothy Gowers, for their support and guidance; the first author is supported by the CB European PhD Studentship funded by Trinity College, Cambridge, and the second
author acknowledges support from the Isaac Newton Trust through a Trinity Cambridge Research Studentship.
Let \((X,\mathcal{A},\mu)\) be a normalized finitely additive measure space, meaning that \(X\) is a set, \(\mathcal{A}\) is an algebra of subsets of \(X\), and \(\mu:\mathcal{A}\to[0,1]\) is a finitely additive function satisfying \(\mu(X)=1\). Given a set \(P\in \mathcal{A}\) with \(\mu(P)>0\), we denote the normalized average of a bounded, \(\mathcal{A}\)-measurable function \(h:X\rightarrow\mathbb{C}\) over \(P\) by \[\mathbb{E}_P[h]=\frac{1}{\mu(P)}\int_P h\,d\mu.\]
In what follows, all partitions considered are finite partitions into measurable sets of positive measure, meaning a partition \(\mathcal{P}\) of \(X\) is a finite collection of disjoint sets \(\{P_1, \dots, P_n\} \subset \mathcal{A}\) such that \(\bigcup_{i=1}^n P_i = X\) and \(\mu(P_i) > 0\) for all \(i \in \{1, \dots, n\}\). Let \(\mathcal{P}\) and \(\mathcal{Q}\) be two partitions of \(X\). We say that \(\mathcal{Q}\) refines \(\mathcal{P}\) if, for every \(Q \in \mathcal{Q}\), there exists some \(P \in \mathcal{P}\) such that \(Q \subseteq P\). Given a part \(P \in \mathcal{P}\), the partition of \(P\) induced by \(\mathcal{Q}\), denoted by \(\mathcal{Q}|_P\), is defined as the collection of sets \(\mathcal{Q}|_P = \{Q \in \mathcal{Q} \mid Q \subseteq P\}\).
For every part \(P \in \mathcal{P}\) of a partition \(\mathcal{P}\), fix a non-empty family of measurable functions \(\mathcal{F}_P \subseteq \{f : P \to \mathbb{C} \mid \|f\|_\infty \le 1\}\). Given an \(\mathcal{A}\)-measurable function \(g: X \to [0, 1]\), we say that \(\mathcal{P}\) is \(\varepsilon\)-regular for \(g\) with respect to the families \((\mathcal{F}_P)_{P\in\mathcal{P}}\) whenever the following inequality holds \[\sum_{P \in \mathcal{P}} \mu(P) \sup_{f \in \mathcal{F}_P} \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f \right] \right| \le \varepsilon,\] where \(g|_P\) represents the restriction of \(g\) to the part \(P\). For notational simplicity, we define the irregularity of \(g\) on \(P\) as \[\operatorname{irr}_g(P) = \sup_{f \in \mathcal{F}_P} \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f \right] \right|.\]
For every positive integer \(M\), we choose a subclass of the partitions of \(X\) with at most \(M\) parts. Its elements will be called admissible partitions, and the subclass will be denoted by \(\mathsf{Part}_M(X)\). A refinement scheme consists of a growth function \(\Phi : \mathbb{N} \to \mathbb{N}\) and a refinement function \(\psi\) that, given a partition \(\mathcal{P} \in \mathsf{Part}_M(X)\) and a family of witnesses \(\{f_P\}_{P \in \mathcal{P}}\) with \(f_P \in \mathcal{F}_P\) for each \(P \in \mathcal{P}\), returns a refinement \(\mathcal{P}' = \psi(\mathcal{P}, \{f_P\}_{P \in \mathcal{P}}) \in \mathsf{Part}_{\Phi(M)}(X)\) of \(\mathcal{P}\).
In this work, we assume that the chosen witnesses become measurable with respect to the refinement, i.e., if \(\mathcal{P}' = \psi(\mathcal{P}, \{f_P\}_{P \in \mathcal{P}})\), then for each \(P \in \mathcal{P}\), the function \(f_P\) is constant on every part of the restricted partition \(\mathcal{P}'|_P\).
This is the distinctive feature of our framework. By isolating the measurability of the chosen witnesses with respect to the refinement as the essential hypothesis, one obtains an abstract regularity lemma that is at once simple, yet general.
With this notation in hand, we can state the unified abstract regularity lemma.
Theorem 1 (Unified abstract regularity lemma). Let \((X, \mathcal{A}, \mu)\) be a normalized finitely additive measure space. Let \(g: X \to [0, 1]\) be an \(\mathcal{A}\)-measurable function, let \(\varepsilon > 0\), and let \(\mathcal{P}_0 \in \mathsf{Part}_{M_0}(X)\). Then there exists an integer \(t \le \lceil \varepsilon^{-2} \rceil\) and a partition \(\mathcal{P}_t \in \mathsf{Part}_{\Phi^{\circ t}(M_0)}(X)\) which is \(\varepsilon\)-regular for \(g\), where \(\Phi^{\circ t}\) denotes the \(t\)-fold composition of \(\Phi\) with itself.
Proof. For a partition \(\mathcal{P}\), define the energy of \(g\) by \[\mathcal{E}_g(\mathcal{P}) = \|\mathbb{E}[g \mid \mathcal{P}]\|_{L^2(\mu)}^2 = \sum_{P \in \mathcal{P}} \mu(P) \bigl(\mathbb{E}_P[g|_P]\bigr)^2.\]
The energy always lies between \(0\) and \(1\), since \(0 \le \mathcal{E}_g(\mathcal{P}) \le \mathbb{E}_\mu[g^2] \le 1\). We first record the energy increment estimate.
Claim. Let \(\mathcal{Q}\) be a partition that refines \(\mathcal{P}\). Suppose that, for each \(P \in \mathcal{P}\), the function \(f_P : P \to \mathbb{C}\) satisfies \(\|f_P\|_\infty \le 1\) and is constant on every part of \(\mathcal{Q}|_P\). Then \[\mathcal{E}_g(\mathcal{Q}) - \mathcal{E}_g(\mathcal{P}) \ge \sum_{P \in \mathcal{P}} \mu(P) \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] \right|^2.\]
Proof of the claim. Since \(\mathcal{Q}\) refines \(\mathcal{P}\), we can write \[\mathcal{E}_g(\mathcal{Q}) = \sum_{P \in \mathcal{P}} \mu(P) \|\mathbb{E}_P[g|_P \mid \mathcal{Q}|_P]\|_{L^2(P)}^2.\] For \(P \in \mathcal{P}\), define \(h_P = \mathbb{E}_P[g|_P \mid \mathcal{Q}|_P] - \mathbb{E}_P[g|_P]\). Then \(\mathbb{E}_P[h_P] = 0\), and hence \[\|\mathbb{E}_P[g|_P \mid \mathcal{Q}|_P]\|_{L^2(P)}^2 = \bigl(\mathbb{E}_P[g|_P]\bigr)^2 + \|h_P\|_{L^2(P)}^2.\] Summing up, we obtain \[\mathcal{E}_g(\mathcal{Q}) - \mathcal{E}_g(\mathcal{P}) = \sum_{P \in \mathcal{P}} \mu(P) \|h_P\|_{L^2(P)}^2.\] Because \(f_P\) is constant on the parts of \(\mathcal{Q}|_P\), we have \[\mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] = \mathbb{E}_P[h_P f_P].\] By the Cauchy–Schwarz inequality, \[\left|\mathbb{E}_P[h_P f_P]\right|^2 \le \mathbb{E}_P[|h_P|^2]\,\mathbb{E}_P[|f_P|^2] \le \|h_P\|_{L^2(P)}^2.\] Multiplying by \(\mu(P)\) and summing over all \(P \in \mathcal{P}\) proves the claim. \(\square\)
We now run the energy increment procedure. Starting from \(\mathcal{P}_0\), suppose that \(\mathcal{P}_s\) has been constructed. If \(\mathcal{P}_s\) is \(\varepsilon\)-regular for \(g\), we stop. Otherwise, \[\sum_{P \in \mathcal{P}_s} \mu(P) \sup_{f \in \mathcal{F}_P} \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f \right] \right| > \varepsilon.\] Since \(\mathcal{P}_s\) is finite, we may choose witnesses \(f_P \in \mathcal{F}_P\) so that \[\sum_{P \in \mathcal{P}_s} \mu(P) \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] \right| > \varepsilon.\] Set \(\mathcal{P}_{s+1} = \psi(\mathcal{P}_s, \{f_P\}_{P \in \mathcal{P}_s})\). By the defining property of the refinement scheme, each \(f_P\) is constant on the parts of \(\mathcal{P}_{s+1}|_P\). The claim then gives \[\mathcal{E}_g(\mathcal{P}_{s+1}) - \mathcal{E}_g(\mathcal{P}_s) \ge \sum_{P \in \mathcal{P}_s} \mu(P) \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] \right|^2.\] Since \(\sum_{P \in \mathcal{P}_s} \mu(P) = 1\), the Cauchy-Schwarz inequality yields \[\sum_{P \in \mathcal{P}_s} \mu(P) \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] \right|^2 \ge \left( \sum_{P \in \mathcal{P}_s} \mu(P) \left| \mathbb{E}_P \left[ (g|_P - \mathbb{E}_P[g|_P]) \cdot f_P \right] \right| \right)^2 > \varepsilon^2.\] Thus, each non-regular step increases the energy by more than \(\varepsilon^2\). Since the energy is bounded above by \(1\), the procedure stops after at most \(t \le \lceil \varepsilon^{-2} \rceil\) steps. Applying the growth function at each step gives the bound \(\mathcal{P}_t \in \mathsf{Part}_{\Phi^{\circ t}(M_0)}(X)\), with a final cardinality at most \(\Phi^{\circ \lceil \varepsilon^{-2} \rceil}(M_0)\). ◻
Fix a prime \(p\) and let \(X=\mathbb{F}_p^n\), equipped with the uniform probability measure. For \(\xi,x\in\mathbb{F}_p^n\), set \(\langle \xi,x\rangle=\sum_{j=1}^n \xi_jx_j\in\mathbb{F}_p\) and define \(\chi_\xi(x)=\omega^{\langle \xi,x\rangle}\), where \(\omega=e^{2\pi i/p}\). Let \(A\subseteq \mathbb{F}_p^n\), and apply 1 to the function \(g=\mathbf{1}_A\).
If \(H\le \mathbb{F}_p^n\) is a subspace, write \(\mathcal{P}_H\) for the partition of \(\mathbb{F}_p^n\) into cosets of \(H\), and write \[H^\perp =\{\xi\in\mathbb{F}_p^n:\langle \xi,h\rangle=0 \text{ for every }h\in H\}.\] For a coset \(P=a+H\), we identify \(P\) with \(H\) by translation. We then take as local test functions \[\mathcal{F}_{a+H} = \{0\} \cup \left\{ x\mapsto \overline{\chi_\xi(x-a)}: \xi\in\mathbb{F}_p^n\setminus H^\perp \right\}.\] The irregularity of \(A\) on the coset \(a+H\) is exactly the largest non-trivial Fourier coefficient of the balanced restriction of \(\mathbf{1}_A\) to that coset \[\operatorname{irr}_{\mathbf{1}_A}(a+H) = \max_{\xi\in\mathbb{F}_p^n\setminus H^\perp} \left| \mathop{\mathbb{E}}\limits_{x\in a+H} \left[ \left(\mathbf{1}_A(x)-\mathop{\mathbb{E}}\limits_{y\in a+H}\mathbf{1}_A(y)\right) \overline{\chi_\xi(x-a)} \right] \right|,\] with the convention that this maximum is \(0\) when \(H=\{0\}\).
The refinement scheme is the usual Fourier refinement. Suppose that, for each coset \(a+H\), we have chosen a witness frequency \(\xi_{a+H}\in\mathbb{F}_p^n\setminus H^\perp\), where the zero witness imposes no condition. Define \[H' = \left\{ h\in H: \chi_{\xi_{a+H}}(h)=1 \text{ for every chosen non-zero witness }\xi_{a+H} \right\}.\] Then \(\mathcal{P}_{H'}\) refines \(\mathcal{P}_H\). Moreover, every chosen witness is constant on each coset of \(H'\) inside its ambient coset \(a+H\). Therefore, the measurability condition required in 1 is satisfied.
It remains only to record the growth of the refinement. If \(\operatorname{codim} H=r\), then \(\mathcal{P}_H\) has \(p^r\) parts, and the above construction adds at most one independent linear condition for each coset of \(H\). Hence, \(\operatorname{codim} H'\le r+p^r.\) Equivalently, if \(\mathcal{P}_H\) has at most \(M\) parts, then \(\mathcal{P}_{H'}\) has at most \(p^{\lceil \log_p M\rceil+M}\) parts. This gives an admissible refinement scheme with a growth function depending only on \(p\).
We now recover the usual form of Green’s arithmetic regularity lemma.
Corollary 2 (Green’s arithmetic regularity lemma [2]). For every prime \(p\) and every \(\varepsilon>0\), there is an integer \(M=M(p,\varepsilon)\) such that the following holds. For every \(n\) and every \(A\subseteq\mathbb{F}_p^n\), there is a subspace \(H\le\mathbb{F}_p^n\) of codimension at most \(M\) such that all but at most an \(\varepsilon\)-fraction of the cosets \(a+H\) satisfy \[\max_{\xi\in\mathbb{F}_p^n\setminus H^\perp} \left| \mathop{\mathbb{E}}\limits_{x\in a+H} \left[ \left(\mathbf{1}_A(x)-\mathop{\mathbb{E}}\limits_{y\in a+H}\mathbf{1}_A(y)\right) \overline{\chi_\xi(x-a)} \right] \right| \le \varepsilon .\]
Proof. Start from the trivial coset partition \(\mathcal{P}_{\mathbb{F}_p^n}\). The refinement scheme described above stays inside the class of coset partitions. Hence, 1 yields a subspace \(H\le \mathbb{F}_p^n\), of codimension bounded only in terms of \(p\) and \(\eta\), such that the partition \(\mathcal{P}_H\) is \(\eta\)-regular for \(\mathbf{1}_A\). Unwinding the definition of regularity, we obtain the following result. For every prime \(p\) and every \(\eta>0\), there is an integer \(M=M(p,\eta)\) such that the following holds. For every \(n\) and every \(A\subseteq \mathbb{F}_p^n\), there is a subspace \(H\le \mathbb{F}_p^n\) of codimension at most \(M\) such that \[\mathop{\mathbb{E}}\limits_{a\in \mathbb{F}_p^n/H} \max_{\xi\in\mathbb{F}_p^n\setminus H^\perp} \left| \mathop{\mathbb{E}}\limits_{x\in a+H} \left[ \left(\mathbf{1}_A(x)-\mathop{\mathbb{E}}\limits_{y\in a+H}\mathbf{1}_A(y)\right) \overline{\chi_\xi(x-a)} \right] \right| \le \eta .\] Taking \(\eta=\varepsilon^2\), it follows that \[\mathop{\mathbb{E}}\limits_{a\in \mathbb{F}_p^n/H} \operatorname{irr}_{\mathbf{1}_A}(a+H) \le \varepsilon^2.\] By Markov’s inequality, the fraction of cosets \(a+H\) for which \(\operatorname{irr}_{\mathbf{1}_A}(a+H)>\varepsilon\) is at most \(\varepsilon\). This is exactly the desired conclusion. ◻
Let \(G=(V,E)\) be a finite simple graph and write \(n=|V|\). We work on \(X=V\times V\) with the uniform probability measure, and we apply Theorem 1 to the function \(g(x,y)=\mathbf{1}_E(x,y)\), where edges are counted as ordered pairs. Thus \(g(x,y)=1\) if \(xy\in E\), and \(g(x,y)=0\) otherwise. The diagonal is irrelevant for the argument. If \(\mathcal{V}=\{V_1,\dots,V_m\}\) is a partition of \(V\), let \(\mathcal{V}^2=\{V_i\times V_j:1\le i,j\le m\}\) be the induced product partition of \(V\times V\). For non-empty subsets \(A,B\subseteq V\), define \(e(A,B)=\abs*{\{(a,b)\in A\times B:ab\in E\}}\), and \(d(A,B)=\frac{e(A,B)}{|A||B|}\). On the part \(V_i\times V_j\), we take as local test functions all rectangle indicators: \[\mathcal{F}_{V_i\times V_j} = \{\mathbf{1}_{S\times T}:S\subseteq V_i,\;T\subseteq V_j\}.\] For this choice, the irregularity on \(V_i\times V_j\) is \[\operatorname{irr}_g(V_i\times V_j) = \frac{1}{|V_i||V_j|} \max_{S\subseteq V_i,\;T\subseteq V_j} \abs*{ e(S,T)-d(V_i,V_j)|S||T| }.\] Indeed, \[\mathbb{E}_{V_i\times V_j} \left[ (g-d(V_i,V_j))\mathbf{1}_{S\times T} \right] = \frac{ e(S,T)-d(V_i,V_j)|S||T| }{ |V_i||V_j| }.\] Therefore the \(\eta\)-regularity condition for the product partition \(\mathcal{V}^2\) becomes \[\sum_{i,j} \max_{S\subseteq V_i,\;T\subseteq V_j} \abs*{ e(S,T)-d(V_i,V_j)|S||T| } \le \eta n^2.\]
The refinement scheme is the usual common refinement. Suppose that, for every ordered pair \((i,j)\), a witness rectangle \(S_{ij}\times T_{ij}\subseteq V_i\times V_j\) has been chosen. We refine each vertex class \(V_i\) by all sets \(S_{ij}\) with \(1\le j\le m\), and \(T_{ji}\) with \(1\le j\le m\). Equivalently, we replace \(V_i\) by the parts of the Boolean algebra generated inside \(V_i\) by these at most \(2m\) subsets. The resulting vertex partition \(\mathcal{V}'\) refines \(\mathcal{V}\), and the product partition \((\mathcal{V}')^2\) refines \(\mathcal{V}^2\). Moreover, every chosen rectangle indicator \(\mathbf{1}_{S_{ij}\times T_{ij}}\) is constant on each part of \((\mathcal{V}')^2|_{V_i\times V_j}\), exactly as required by the abstract refinement scheme.
If \(\mathcal{V}\) has \(m\) parts, then each \(V_i\) is split into at most \(2^{2m}\) parts. Hence \(\mathcal{V}'\) has at most \(m2^{2m}\) parts, and the product partition has at most \(m^2 2^{4m}\) parts. Thus, this is an admissible refinement scheme with a growth function depending only on the current number of parts.
We now recover the usual form of Szemerédi’s graph regularity lemma.
Corollary 3 (Szemerédi’s graph regularity lemma [1]). For every \(\varepsilon>0\), there is an integer \(M=M(\varepsilon)\) such that every finite graph \(G=(V,E)\) admits a partition \(V=V_1\cup\cdots\cup V_m\) with \(1\le m\le M\) such that \[\sum_{\substack{1\le i,j\le m\\ (V_i,V_j)\text{ is not }\varepsilon\text{-regular}}} |V_i||V_j| \le \varepsilon|V|^2.\]
Proof. Start from the trivial partition \(V_1=V\) and apply 1 to the induced product partition of \(V\times V\), with the rectangle test families and refinement scheme described above. The final product partition is induced by a vertex partition \(\mathcal{V}=\{V_1,\dots,V_m\}\), where \(m\) is bounded only in terms of \(\eta\). Unwinding the definition of abstract regularity, we obtain the following result. For every \(\eta>0\), there is an integer \(M=M(\eta)\) such that every finite graph \(G=(V,E)\) admits a partition \(V=V_1\cup\cdots\cup V_m\), \(1\le m\le M\), satisfying \[\sum_{i,j} \max_{S\subseteq V_i,\;T\subseteq V_j} \abs*{ e(S,T)-d(V_i,V_j)|S||T| } \le \eta |V|^2.\] We now relate this averaged formulation to the usual pairwise notion of regularity. Recall that a pair \((A,B)\) is \(\varepsilon\)-regular if, for all \(S\subseteq A\) and \(T\subseteq B\) with \(|S|\ge \varepsilon|A|\), \(|T|\ge \varepsilon|B|\), one has \(\abs*{d(S,T)-d(A,B)}\le \varepsilon.\) If \((V_i,V_j)\) is not \(\varepsilon\)-regular, then there are \(S\subseteq V_i\) and \(T\subseteq V_j\) such that \(|S|\ge \varepsilon|V_i|\), \(|T|\ge \varepsilon|V_j|\), and \(\abs*{d(S,T)-d(V_i,V_j)}>\varepsilon\). Equivalently, \[\abs*{ e(S,T)-d(V_i,V_j)|S||T| } > \varepsilon|S||T| \ge \varepsilon^3 |V_i||V_j|.\] Let \(\eta=\varepsilon^4\). Let \(\mathcal{B}\) be the set of ordered pairs \((i,j)\) for which \((V_i,V_j)\) is not \(\varepsilon\)-regular. We have that \[\varepsilon^3 \sum_{(i,j)\in\mathcal{B}} |V_i||V_j| \le \sum_{i,j} \max_{S\subseteq V_i,\;T\subseteq V_j} \abs*{ e(S,T)-d(V_i,V_j)|S||T| } \le \varepsilon^4 |V|^2.\] Dividing by \(\varepsilon^3\) and recalling that \(\eta=\varepsilon^4\), we obtain \[\sum_{\substack{1\le i,j\le m\\ (V_i,V_j)\text{ is not }\varepsilon\text{-regular}}} |V_i||V_j| \le \varepsilon|V|^2.\] ◻
For a finite set \(I\) and \(p\in(0,1)\), let \(\mu_p^I\) denote the \(p\)-biased product measure on \(\{0,1\}^I\), that is, \[\mu_p^I(\{z\})=\prod_{i\in I}p^{z_i}(1-p)^{1-z_i}.\] When the set of coordinates is clear, we simply write \(\mu_p\).
If \(f:\{0,1\}^n\to[0,1]\), we write \(\mu_p(f)=\mathbb{E}_{x\sim\mu_p^{[n]}}f(x)\). If \(T\subseteq[n]\) and \(z\in\{0,1\}^T\), then \(f_{T\to z}\) denotes the restriction of \(f\) obtained by fixing the coordinates in \(T\) according to \(z\).
Let \(r\in\mathbb{N}\) and \(\varepsilon>0\). A function \(f:\{0,1\}^n\to[0,1]\) is called \((r,\varepsilon)\)-quasirandom with respect to \(p\) if, for every \(R\subseteq[n]\) with \(|R|\le r\) and every \(w\in\{0,1\}^R\),\[\abs*{\mu_p(f_{R\to w})-\mu_p(f)}\le \varepsilon.\] More generally, if \(T\subseteq[n]\) and \(z\in\{0,1\}^T\), we say that \(f_{T\to z}\) is \((r,\varepsilon)\)-quasirandom if, for every \(R\subseteq[n]\setminus T\) with \(|R|\le r\) and every \(w\in\{0,1\}^R\), \[\abs*{\mu_p(f_{T\to z,R\to w})-\mu_p(f_{T\to z})}\le \varepsilon.\]
Fix \(p\in(0,1)\) and put \(X=\{0,1\}^n\), \(\mu=\mu_p^{[n]}\), \(g=f\). For \(T\subseteq[n]\), let \(\mathcal{P}_T\) be the partition of \(\{0,1\}^n\) into the parts \(P_z=\{x\in\{0,1\}^n:x_T=z\}\) with \(z\in\{0,1\}^T.\) On the part \(P_z\) we take as test functions all indicators of subcubes obtained by fixing at most \(r\) further coordinates: \[\mathcal{F}_{P_z} = \{0\} \cup \Bigl\{ \mathbf{1}_{\{x_R=w\}}\bigm|_{P_z}: R\subseteq[n]\setminus T,\;|R|\le r,\;w\in\{0,1\}^R \Bigr\}.\] Here \(\mathbf{1}_{\{x_R=w\}}\bigm|_{P_z}\) is viewed as a function on \(P_z\). For such a test function we have \[\mathbb{E}_{P_z} \Bigl[ \bigl(g-\mathbb{E}_{P_z}g\bigr)\mathbf{1}_{\{x_R=w\}} \Bigr] = \mu_p^R(w) \Bigl( \mu_p(f_{T\to z,R\to w})-\mu_p(f_{T\to z}) \Bigr).\] Indeed, after conditioning on \(x_T=z\), the remaining coordinates still have the \(p\)-biased product distribution. Therefore \[\operatorname{irr}_g(P_z) = \sup_{\substack{R\subseteq[n]\setminus T\\ |R|\le r}} \;\sup_{w\in\{0,1\}^R} \mu_p^R(w) \abs*{ \mu_p(f_{T\to z,R\to w})-\mu_p(f_{T\to z}) }.\]
The refinement scheme is also immediate. Suppose that, for each part \(P_z\), a witness has been chosen. If the witness is non-zero, it is of the form \(\mathbf{1}_{\{x_{R_z}=w_z\}}\) with \(R_z\subseteq[n]\setminus T\) and \(|R_z|\le r\). Define \[T' = T\cup\bigcup_{z\in\{0,1\}^T}R_z,\] where zero witnesses contribute nothing. Then \(\mathcal{P}_{T'}\) refines \(\mathcal{P}_T\), and every chosen witness is constant on every part of \(\mathcal{P}_{T'}\) contained in the corresponding part of \(\mathcal{P}_T\). Moreover, \(|T'|\le |T|+r2^{|T|}\). If one parametrizes admissible partitions by their number of parts, this gives the growth bound \(\Phi_r(M)=\left\lceil M2^{rM}\right\rceil,\) because a partition fixing at most \(\lfloor\log_2 M\rfloor\) coordinates has at most \(M\) parts, and after refinement it has at most \(M2^{rM}\) parts.
Corollary 4 (Regularity lemma for Boolean functions). For every \(r\in\mathbb{N}\) and every \(\varepsilon,\zeta,\delta>0\), there is an integer \(J=J(r,\varepsilon,\zeta,\delta)\) such that the following holds. Let \(\zeta<p<\frac{1}{2}-\zeta\), and let \(f:\{0,1\}^n\to\{0,1\}\). Then there is a set of coordinates \(T\subseteq[n]\) with \(|T|\le J\) such that \[\Pr_{z\sim\mu_p^T} \left[ f_{T\to z}\text{ is not }(r,\varepsilon)\text{-quasirandom with respect to }p \right] \le \delta.\]
Proof. Apply 1 to the partitions \(\mathcal{P}_T\) and to the test families described above, with regularity parameter \(\eta=\delta\,\zeta^r\varepsilon\). This gives a coordinate set \(T\) of size at most a constant \(J=J(r,\varepsilon,\zeta,\delta)\), independent of \(n\), such that \(\sum_{z\in\{0,1\}^T} \mu_p^T(z)\operatorname{irr}_f(P_z) \le\eta\).
Let \(B= \{z\in\{0,1\}^T: f_{T\to z}\text{ is not }(r,\varepsilon)\text{-quasirandom}\}\). We claim that every \(z\in B\) satisfies \(\operatorname{irr}_f(P_z)>\zeta^r\varepsilon\). Indeed, if \(z\in B\), then there exist \(R\subseteq[n]\setminus T\) with \(|R|\le r\) and \(w\in\{0,1\}^R\) such that \[\abs*{ \mu_p(f_{T\to z,R\to w})-\mu_p(f_{T\to z}) } > \varepsilon.\] Since \(\zeta<p<\frac{1}{2}-\zeta\), every part of the \(p\)-biased measure on \(R\) has measure at least \(\zeta^{|R|}\ge \zeta^r\). Hence \[\operatorname{irr}_f(P_z) \ge \mu_p^R(w) \abs*{ \mu_p(f_{T\to z,R\to w})-\mu_p(f_{T\to z}) } > \zeta^r\varepsilon.\]
Therefore \[\delta\zeta^r\varepsilon = \eta \ge \sum_{z\in\{0,1\}^T} \mu_p^T(z)\operatorname{irr}_f(P_z) \ge \sum_{z\in B} \mu_p^T(z)\operatorname{irr}_f(P_z) > \mu_p^T(B)\zeta^r\varepsilon.\] Cancelling \(\zeta^r\varepsilon\) gives \(\mu_p^T(B)\le\delta\), which is the desired conclusion. ◻
We first recall the setup. Let \((X,\mathcal{A},\mu,\mathcal{S})\) be an SR-system in the sense of Bollobás and Nikiforov: \((X,\mathcal{A},\mu)\) is a normalized measure triple and \(\mathcal{S}\subseteq\mathcal{A}\) is a semiring. Further, assume that there is an integer \(r\ge 1\) such that, whenever \(S,T\in\mathcal{S}\), the difference \(S\setminus T\) can be written as a disjoint union of at most \(r\) members of \(\mathcal{S}\). If \(A,V\in\mathcal{A}\) and \(\mu(V)>0\), write \(d(A,V)=\frac{\mu(A\cap V)}{\mu(V)}.\)
Following Bollobás and Nikiforov, if \(V\in\mathcal{S}\) has positive measure, we say that \(A\) is \(\varepsilon\)-regular in \(V\) if \[\abs*{d(A,U)-d(A,V)}<\varepsilon,\] for every \(U\in\mathcal{S}\) with \(U\subseteq V\) and \(\mu(U)>\varepsilon\mu(V)\). If \(\mathcal{P}\) is a finite partition of \(X\) into members of \(\mathcal{S}\), we say that \(A\) is \(\varepsilon\)-regular in \(\mathcal{P}\) if \[\sum_{\substack{P\in\mathcal{P}\\ A\text{ is not }\varepsilon\text{-regular in }P}} \mu(P)<\varepsilon.\]
We shall also use the notion of a bounding family of partitions. Let \(\Pi(\mathcal{S})\) denote the set of finite partitions of \(X\) into members of \(\mathcal{S}\). A family \(\Phi\subseteq \Pi(\mathcal{S})\) bounds \(\Pi(\mathcal{S})\) if there is an increasing function \(\varphi:\mathbb{N}\to\mathbb{N}\) such that, for every \(\mathcal{P}\in\Pi(\mathcal{S})\), there exists \(\mathcal{Q}\in\Phi\) refining \(\mathcal{P}\) with \(|\mathcal{Q}|\le \varphi(|\mathcal{P}|)\).
We now recover the abstract regularity lemma of Bollobás and Nikiforov.
Corollary 5 (Bollobás–Nikiforov abstract regularity lemma). Let \((X,\mathcal{A},\mu,\mathcal{S})\) be an SR-system, and let \(\Phi\subseteq\Pi(\mathcal{S})\) be a family of partitions bounding \(\Pi(\mathcal{S})\) with rate \(\varphi\). Let \(\mathcal{L}\subseteq\mathcal{A}\) be a finite family of measurable sets, let \(\mathcal{P}\in\Pi(\mathcal{S})\), and let \(\varepsilon>0\). Then there is an integer \(q=q(\varepsilon,|\mathcal{L}|,|\mathcal{P}|)\) and a partition \(\mathcal{Q}\in\Phi\) such that \(\mathcal{Q}\) refines \(\mathcal{P}\), \(|\mathcal{Q}|\le q,\) and every \(A\in\mathcal{L}\) is \(\varepsilon\)-regular in \(\mathcal{Q}\).
Proof. Let \(\ell=|\mathcal{L}|\). If \(\ell=0\) there is nothing to prove, so assume \(\ell\ge 1\). Since \(\Phi\) bounds \(\Pi(\mathcal{S})\), choose first a partition \(\mathcal{P}_0\in\Phi\) refining \(\mathcal{P}\) with \(|\mathcal{P}_0|\le \varphi(|\mathcal{P}|)\). We apply the proof of 1 simultaneously to the functions \(\mathbf{1}_A\), \(A\in\mathcal{L}\). For a part \(P\) of a partition \(\mathcal{R}\in\Phi\), take as local test functions \[\mathcal{F}_P = \{0\}\cup \{\mathbf{1}_U|_P: U\in\mathcal{S},\;U\subseteq P\}.\] For \(A\in\mathcal{L}\) and \(P\in\mathcal{R}\), the corresponding local irregularity is \[\operatorname{irr}_A(P) = \sup_{\substack{U\in\mathcal{S}\\ U\subseteq P}} \left| \mathbb{E}_P\left[ \bigl(\mathbf{1}_A-d(A,P)\bigr)\mathbf{1}_U \right] \right| = \sup_{\substack{U\in\mathcal{S}\\ U\subseteq P}} \frac{\mu(U)}{\mu(P)} \abs*{d(A,U)-d(A,P)}.\]
The only point to check is that the refinement scheme required by 1 is available inside the family \(\Phi\). Suppose that \(\mathcal{R}\in\Phi\) has \(m\) parts. For each pair \((A,P)\in\mathcal{L}\times\mathcal{R}\), choose a witness \(U_{A,P}\in\mathcal{S}\) with \(U_{A,P}\subseteq P\). Inside a fixed part \(P\), we refine by all the sets \(U_{A,P}\), \(A\in\mathcal{L}\). Since \(\mathcal{S}\) is \(r\)-built, refining one member of \(\mathcal{S}\) by one semiring subset splits it into at most \(r+1\) members of \(\mathcal{S}\). Iterating this for the \(\ell\) witnesses inside \(P\), we obtain a partition of \(P\) into at most \((r+1)^\ell\) members of \(\mathcal{S}\) on which all the indicators \(\mathbf{1}_{U_{A,P}}\) are constant. Doing this for every \(P\in\mathcal{R}\) gives a partition \(\mathcal{R}^\ast\in\Pi(\mathcal{S})\) with \(|\mathcal{R}^\ast|\le m(r+1)^\ell\). Now use the bounding property of \(\Phi\) to choose \(\mathcal{R}'\in\Phi\) refining \(\mathcal{R}^\ast\) with \[|\mathcal{R}'|\le \varphi\bigl(m(r+1)^\ell\bigr).\] Thus we have an admissible refinement scheme with growth function \[\Psi(m)=\varphi\bigl(m(r+1)^\ell\bigr).\] Run the energy-increment argument with the total energy \[\mathcal{E}_{\mathcal{L}}(\mathcal{R}) = \sum_{A\in\mathcal{L}} \left\| \mathbb{E}[\mathbf{1}_A\mid \mathcal{R}] \right\|_{L^2(\mu)}^2.\] This energy lies between \(0\) and \(\ell\). If \[\sum_{A\in\mathcal{L}} \sum_{P\in\mathcal{R}} \mu(P)\operatorname{irr}_A(P)>\eta,\] then choose witnesses \(U_{A,P}\) and refine as above. The same computation as in the proof of 1 gives \[\mathcal{E}_{\mathcal{L}}(\mathcal{R}') - \mathcal{E}_{\mathcal{L}}(\mathcal{R}) \ge \sum_{A\in\mathcal{L}} \sum_{P\in\mathcal{R}} \mu(P) \left| \mathbb{E}_P\left[ \bigl(\mathbf{1}_A-d(A,P)\bigr)\mathbf{1}_{U_{A,P}} \right] \right|^2 .\] Since the weights \(\mu(P)\) sum to \(1\) for each fixed \(A\), and there are \(\ell\) choices of \(A\), Jensen’s inequality gives \[\sum_{A\in\mathcal{L}} \sum_{P\in\mathcal{R}} \mu(P) \left| \mathbb{E}_P\left[ \bigl(\mathbf{1}_A-d(A,P)\bigr)\mathbf{1}_{U_{A,P}} \right] \right|^2 \ge \frac{\eta^2}{\ell}.\] Thus every non-regular step increases the total energy by at least \(\eta^2/\ell\). Since the total energy is at most \(\ell\), the procedure stops after at most \(\left\lceil \frac{\ell^2}{\eta^2}\right\rceil\) steps. We obtain a partition \(\mathcal{Q}\in\Phi\) refining \(\mathcal{P}_0\), hence refining \(\mathcal{P}\), with \(|\mathcal{Q}| \le \Psi^{\circ \left\lceil \frac{\ell^2}{\eta^2}\right\rceil}\bigl(\varphi(|\mathcal{P}|)\bigr),\) and satisfying \[\sum_{A\in\mathcal{L}} \sum_{Q\in\mathcal{Q}} \mu(Q)\operatorname{irr}_A(Q) \le \eta .\] It remains to translate this averaged test-function regularity into the Bollobás–Nikiforov notion. Fix \(A\in\mathcal{L}\), and let \(\mathcal{B}_A\) be the set of parts \(Q\in\mathcal{Q}\) in which \(A\) is not \(\varepsilon\)-regular. For each \(Q\in\mathcal{B}_A\), there is a set \(U\in\mathcal{S}\), \(U\subseteq Q\), such that \(\mu(U)>\varepsilon\mu(Q)\) and \(\abs*{d(A,U)-d(A,Q)}>\varepsilon\). Therefore \[\operatorname{irr}_A(Q) \ge \frac{\mu(U)}{\mu(Q)} \abs*{d(A,U)-d(A,Q)} > \varepsilon^2 .\] Hence \[\varepsilon^2 \sum_{Q\in\mathcal{B}_A}\mu(Q) < \sum_{Q\in\mathcal{B}_A} \mu(Q)\operatorname{irr}_A(Q) \le \sum_{Q\in\mathcal{Q}} \mu(Q)\operatorname{irr}_A(Q) \le \eta .\] Taking \(\eta=\varepsilon^3\) gives \(\sum_{Q\in\mathcal{B}_A}\mu(Q)<\varepsilon\). Thus every \(A\in\mathcal{L}\) is \(\varepsilon\)-regular in \(\mathcal{Q}\). Finally, with \(\eta=\varepsilon^3\), one may take \(q(\varepsilon,\ell,|\mathcal{P}|) = \Psi^{\circ \left\lceil \ell^2\varepsilon^{-6}\right\rceil} \bigl(\varphi(|\mathcal{P}|)\bigr)\), and \(\Psi(m)=\varphi\bigl(m(r+1)^\ell\bigr)\). ◻
We shall use the following harmless variant of 1: the function \(g:X\to[0,1]\) may be replaced by a real-valued function \(X\in L^2(\mu)\) with \(\|X\|_2\le 1\). The proof is unchanged, since the only point at which boundedness is used is the energy bound \[0\le \|\mathbb{E}[X\mid \mathcal{P}]\|_2^2\le \|X\|_2^2\le 1.\]
We identify a finite sub-\(\sigma\)-algebra with its finite partition into parts. If \(\mathcal{B}\) is a finite sub-\(\sigma\)-algebra, write \(\operatorname{complex}(\mathcal{B})\) for the least number of events needed to generate it. Thus \(\mathcal{B}\) has at most \(2^{\operatorname{complex}(\mathcal{B})}\) parts. If \((\mathcal{B}_i)_{i\in I}\) is a finite family of finite sub-\(\sigma\)-algebras, write \(\bigvee_{i\in I}\mathcal{B}_i\) for their common refinement. We first isolate the single-level regularization statement supplied by 1.
Lemma 6. Let \((\Omega,\mathcal{B}_{\max},\mu)\) be a probability space, let \((\mathcal{B}_{i,\max})_{i\in I}\) be a finite family of sub-\(\sigma\)-algebras of \(\mathcal{B}_{\max}\), and let \(X\in L^2(\mathcal{B}_{\max})\) satisfy \(\|X\|_2\le 1\). Let \((\mathcal{B}_i)_{i\in I}\) be finite sub-\(\sigma\)-algebras with \(\mathcal{B}_i\subseteq \mathcal{B}_{i,\max}\) and \(\operatorname{complex}(\mathcal{B}_i)\le M\) for all \(i\in I\). Then, for every \(\delta>0\), there are finite sub-\(\sigma\)-algebras \(\mathcal{B}_i\subseteq \mathcal{B}_i'\subseteq \mathcal{B}_{i,\max}\) such that \[\left| \mathbb{E}\left[ \left( X-\mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i'\right] \right) \prod_{i\in I}1_{A_i} \right] \right| \le \delta\] for every choice of events \(A_i\in\mathcal{B}_{i,\max}\). Moreover, the complexities of the \(\mathcal{B}_i'\) are bounded in terms of \(M,\delta\), and \(|I|\) only.
Proof. Let \(\mathcal{P}=\bigvee_{i\in I}\mathcal{B}_i\) be the joint part partition. For a part \(P\in\mathcal{P}\), take as local test functions \(\mathcal{F}_P = \left\{ \left. \prod_{i\in I}1_{A_i} \right|_P \,:\, A_i\in\mathcal{B}_{i,\max} \right\}.\) These functions are bounded by \(1\).
We now describe the refinement scheme. Suppose that, for each part \(P\in\mathcal{P}\), a witness \(f_P= \left. \prod_{i\in I}1_{A_i(P)} \right|_P\) has been chosen, with \(A_i(P)\in\mathcal{B}_{i,\max}\). Refine each \(\mathcal{B}_i\) by adjoining all the events \(A_i(P)\), as \(P\) ranges over the parts of \(\mathcal{P}\). Denote the resulting algebra by \(\mathcal{B}_i^+\). Then \(\mathcal{B}_i\subseteq \mathcal{B}_i^+\subseteq \mathcal{B}_{i,\max}.\) Moreover, every \(f_P\) is measurable with respect to \(\bigvee_{i\in I}\mathcal{B}_i^+\) inside \(P\), so the measurability condition in 1 is satisfied.
If \(\operatorname{complex}(\mathcal{B}_i)\le M\) for every \(i\), then \(\bigvee_{i\in I}\mathcal{B}_i\) has at most \(2^{|I|M}\) parts. Hence each \(\mathcal{B}_i\) is refined by adjoining at most \(2^{|I|M}\) new events, and so \(\operatorname{complex}(\mathcal{B}_i^+) \le M+2^{|I|M}.\) Thus we have an admissible refinement scheme with a growth function depending only on \(|I|\).
Applying 1 with parameter \(\delta\), starting from \(\bigvee_i\mathcal{B}_i\), gives finite refinements \(\mathcal{B}_i'\supseteq \mathcal{B}_i\) such that, if \(\mathcal{P}'=\bigvee_{i\in I}\mathcal{B}_i',\) then \(\sum_{P\in\mathcal{P}'} \mu(P) \sup_{A_i\in\mathcal{B}_{i,\max}} \left| \mathbb{E}_P\left[ \left(X-\mathbb{E}_P X\right) \prod_{i\in I}1_{A_i} \right] \right| \le \delta.\) Now fix arbitrary \(A_i\in\mathcal{B}_{i,\max}\). Since \(\mathbb{E}[X\mid\mathcal{P}']\) is equal to \(\mathbb{E}_PX\) on each part \(P\in\mathcal{P}'\), we have \[\begin{align} & \left| \mathbb{E}\left[ \left( X-\mathbb{E}[X\mid\mathcal{P}'] \right) \prod_{i\in I}1_{A_i} \right] \right| \le \sum_{P\in\mathcal{P}'} \mu(P) \left| \mathbb{E}_P\left[ \left(X-\mathbb{E}_PX\right) \prod_{i\in I}1_{A_i} \right] \right| \le \delta. \end{align}\] This proves the desired product-test regularity. The complexity bound follows by iterating the growth function for at most \(\lceil\delta^{-2}\rceil\) steps. ◻
We now recover Tao’s theorem.
Corollary 7. Let \((\Omega,\mathcal{B}_{\max},\mu)\) be a probability space, let \((\mathcal{B}_{i,\max})_{i\in I}\) be a finite family of sub-\(\sigma\)-algebras of \(\mathcal{B}_{\max}\), and let \(X\in L^2(\mathcal{B}_{\max})\) satisfy \(\|X\|_2\le 1\). Let \(\varepsilon>0\), let \(m\ge 0\), and let \(F:\mathbb{R}_+\to\mathbb{R}_+\) be increasing. Then there are finite sub-\(\sigma\)-algebras \(\mathcal{B}_i\subseteq\mathcal{B}_i'\subseteq\mathcal{B}_{i,\max}\), \(i\in I\), and a number \(M=O_{\varepsilon,F,m,|I|}(1)\), such that \(M\ge m\), \(\operatorname{complex}(\mathcal{B}_i)\le M\) for every \(i\in I\), \(\left\| \mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i'\right] - \mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i\right] \right\|_2 \le \varepsilon,\) and, for every choice of events \(A_i\in\mathcal{B}_{i,\max}\), \[\left| \mathbb{E}\left[ \left( X-\mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i'\right] \right) \prod_{i\in I}1_{A_i} \right] \right| \le \frac{1}{F(M)}.\]
Proof. We run a coarse/fine energy-increment procedure. Begin for every \(i\in I\), with the trivial algebras \(\mathcal{B}_i^{(0)}=\{\emptyset,\Omega\}\). Suppose that the current coarse algebras \((\mathcal{B}_i^{(s)})_{i\in I}\) have been constructed, and put \(M_s= \max\left\{ m,\max_{i\in I}\operatorname{complex}(\mathcal{B}_i^{(s)}) \right\}.\) Apply 6 to the coarse algebras \((\mathcal{B}_i^{(s)})_{i\in I}\) with \(\delta_s=\frac{1}{F(M_s)}.\) This gives refinements \(\mathcal{B}_i^{(s)}\subseteq \mathcal{B}_i^{\mathrm{fine}} \subseteq \mathcal{B}_{i,\max}\) such that, for every \(A_i\in\mathcal{B}_{i,\max}\), \[\left| \mathbb{E}\left[ \left( X-\mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i^{\mathrm{fine}}\right] \right) \prod_{i\in I}1_{A_i} \right] \right| \le \frac{1}{F(M_s)}.\]
If \(\left\| \mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i^{\mathrm{fine}}\right] - \mathbb{E}\left[X\mid \bigvee_{i\in I}\mathcal{B}_i^{(s)}\right] \right\|_2 \le \varepsilon,\) we stop and set \(\mathcal{B}_i=\mathcal{B}_i^{(s)}, \mathcal{B}_i'=\mathcal{B}_i^{\mathrm{fine}}, M=M_s.\) The required conclusions then hold.
Otherwise, set \(\mathcal{B}_i^{(s+1)}=\mathcal{B}_i^{\mathrm{fine}}\) and continue. We claim that this can happen at most \(\lceil\varepsilon^{-2}\rceil\) times. Indeed, write \(\mathcal{C}_s=\bigvee_{i\in I}\mathcal{B}_i^{(s)}\) and \(\mathcal{C}_s^{\mathrm{fine}} = \bigvee_{i\in I}\mathcal{B}_i^{\mathrm{fine}}.\) Since \(\mathcal{C}_s^{\mathrm{fine}}\) refines \(\mathcal{C}_s\), orthogonality of conditional expectation gives \(\left\|\mathbb{E}[X\mid\mathcal{C}_s^{\mathrm{fine}}]\right\|_2^2 - \left\|\mathbb{E}[X\mid\mathcal{C}_s]\right\|_2^2 = \left\| \mathbb{E}[X\mid\mathcal{C}_s^{\mathrm{fine}}] - \mathbb{E}[X\mid\mathcal{C}_s] \right\|_2^2.\) Thus every non-stopping step increases the energy by more than \(\varepsilon^2\). But the energy is always at most \(\|X\|_2^2\le 1\), so there are at most \(\lceil\varepsilon^{-2}\rceil\) non-stopping steps.
It remains only to bound the final value of \(M\). By 6, the complexity of the fine algebras constructed from coarse complexity at most \(M_s\) is bounded by a quantity depending only on \(M_s\), \(F(M_s)\), and \(|I|\). Hence there is an increasing function \(\Theta=\Theta_{F,|I|}\) such that \(M_{s+1}\le \Theta(M_s)\) whenever the procedure does not stop. Since the procedure has at most \(\lceil\varepsilon^{-2}\rceil\) non-stopping steps and \(M_0=m\), the final value satisfies \(M\le \Theta^{\circ \lceil\varepsilon^{-2}\rceil}(m).\) Therefore \(M=O_{\varepsilon,F,m,|I|}(1)\), as required. ◻