diff --git a/pointwise_multiple_ergodic_mixing_v2.txt b/pointwise_multiple_ergodic_mixing_v2.txt
index 05a9fba..058be02 100644
--- a/pointwise_multiple_ergodic_mixing_v2.txt
+++ b/pointwise_multiple_ergodic_mixing_v2.txt
@@ -11,8 +11,8 @@
 % Reading keys, so nothing is hidden:
 %
 %   (X,F,mu,T)      T is invertible, bimeasurable and mu-preserving.
-%   mixing          mu(A cap T^{-r}B) -> mu(A)mu(B) as |r| -> infty.
-%   L^infty         bounded real measurable functions (complex by linearity).
+%   mixing          mu(A cap T^{-r}B) -> mu(A)mu(B) as |r| -> infinity.
+%   L^infinity         bounded real measurable functions (complex by linearity).
 %   L^2, ip, norm   the usual Hilbert space of mu-square-integrable functions.
 %   A_N             A_N(f_1,...,f_n)(x) = (1/N) sum_{k=1}^N prod_j f_j(T^{jk}x).
 %
@@ -48,7 +48,7 @@ maketitle
 
 Abstract: 
 Let T be an invertible mixing probability-preserving transformation
-of a probability space (X,mathcal F,calF,mu).  For every integer nge 2
+of a probability space (X,F,mu).  For every integer nge 2
 and every tuple f_1,...,f_nin L^∞(mu), the consecutive
 multiple ergodic averages
 [
@@ -56,7 +56,7 @@ multiple ergodic averages
 ]
 converge mu-almost everywhere to prod_{j=1}^nint_X f_jddmu as
 Nto∞ through all positive integers.  No rate of mixing is
-assumed, and (X,mathcal F,calF,mu) need not be a standard probability
+assumed, and (X,F,mu) need not be a standard probability
 space.
 
 
@@ -66,9 +66,9 @@ space.
 ## The problem and the answer
 
 noindentSetting.
-(X,mathcal F,calF,mu) is a probability space, T:Xto X is invertible,
+(X,F,mu) is a probability space, T:Xto X is invertible,
 bimeasurable and measure preserving.  T is mixing if for every
-A,Binmathcal FcalF
+A,BinF
 
  mu(Acap T^{-r}B)→mu(A)mu(B)(|r|to∞).
 
@@ -86,7 +86,7 @@ If T is mixing, then for mu-almost every x
 [
  A_N(f_1,...,f_n)(x)→prod_{j=1}^nint_X f_jddmu
 ]
-as Nto∞ through all positive integers.  The space (X,mathcal F,calF,mu)
+as Nto∞ through all positive integers.  The space (X,F,mu)
 need not be standard and no rate of mixing is assumed.
 
 The exceptional null set depends on the tuple (f_1,...,f_n) and on
@@ -156,7 +156,7 @@ diving into details.
   and the original spatial coordinate has product law jointly with any proper
   collection of groups.  Two facts are needed.  Conditioning can be strengthened
   to conditioning on W_Y, the maximal measure-distal factor.  And the
-  construction Yrightsquigarrowmathsf M_b(Y)sfM_b(Y)
+  construction YrightsquigarrowM_b(Y)
   (each slot gets its own copy of Y, scaled by 1/b) is a channel.
   The second fact is where tensor packing enters; it is the technical heart.
 [(4)] Minimum-support correlation and saturation.
@@ -169,7 +169,7 @@ diving into details.
   parameters together with one slot sigma(P,y_l).  A short repackaging turns
   the support-k correlation on the saturated channel into a correlation
   supported on k slots of the regrouped channel
-  Z=mathsf M_b(Y),sfM_b(Y), where one slot flips from input to output.
+  Z=M_b(Y), where one slot flips from input to output.
 [(5)] Square argument and contradiction.
   On the selected law of Z we have an Itimes I array.  Row b and column
   b both have the old law lambda_Y; they meet at a common corner.  Saturation
@@ -190,7 +190,7 @@ distal systems.
 Ssec:inputs     & standard models, and inputs (A)--(C) stated precisely.
 Ssec:selected   & the failure witness, the suspension, the selectors, the laws lambda_Y.
 Ssec:channels   & channels, channel conditioning, channel closure by tensor packing.
-Ssec:correlations& minimum support, saturation, the regrouped channel Z=mathsf M_b(Y).sfM_b(Y).
+Ssec:correlations& minimum support, saturation, the regrouped channel Z=M_b(Y).
 Ssec:square     & the square argument and the contradiction.
 Ssec:packing    & appendix: the tensor packing theorem.
 
@@ -215,7 +215,7 @@ for any finite number of slots we have only L^2 convergence.
 
 ## Reduction to a standard probability space
 
-Let (X,mathcal F,calF,mu,T) be the fixed system, and f_1,...,f_n bounded
+Let (X,F,mu,T) be the fixed system, and f_1,...,f_n bounded
 real functions.  Record all integer translates of the f_j as coordinates
 in a single map Phi:Xto K^{Ztimes{1,...,n}}, where K=[-M,M] is
 a compact real interval containing the ranges of all f_j.  The pushforward
@@ -227,7 +227,7 @@ The original functions become coordinate functions on this factor.  A
 countable union of translated null sets runs through the change of
 representatives, so mu-almost-everywhere convergence of the pushed-forward
 averages pulls back to the same statement on X.  From now on the
-system is standard: (X,mathcal F,calF,mu,T) is a standard Borel probability
+system is standard: (X,F,mu,T) is a standard Borel probability
 space with an invertible measure-preserving transformation.
 
 Throughout the proof, factors and conditional laws are understood modulo null
@@ -265,14 +265,14 @@ system, not just a mixing one.
 
 [Averaging profile]
 An averaging profile is a real compactly supported smooth function
-varphi on R whose mass is one and whose moments of order 1,2,3 vanish:
+phi on R whose mass is one and whose moments of order 1,2,3 vanish:
 [
- int_Rvarphi(s)dd s=1,
- int_R s^rvarphi(s)dd s=0quad(r=1,2,3).
+ int_Rphi(s)dd s=1,
+ int_R s^rphi(s)dd s=0quad(r=1,2,3).
 ]
-It is standard if, for some constant Gge 1 depending on varphi,
+It is standard if, for some constant Gge 1 depending on phi,
 [
- norm{varphi^{(r)}}_∞le (G(r+2))^{G(r+2)}(rge0).
+ norm{phi^{(r)}}_∞le (G(r+2))^{G(r+2)}(rge0).
 ]
 
 
@@ -281,16 +281,16 @@ It is standard if, for some constant Gge 1 depending on varphi,
 Let R be any invertible probability-preserving transformation of a
 standard probability space.  Let b_1,b_2,b_3 be distinct nonzero integers
 with greatest common divisor one, and let g_1,g_2,g_3 be bounded real
-functions.  Then for every standard averaging profile varphi,
+functions.  Then for every standard averaging profile phi,
 [
- V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)prod_{j=1}^3 g_j(R^{b_jl}y)
+ V_k^phi(y)=sum_{linZ}2^{-k}phi(l/2^k)prod_{j=1}^3 g_j(R^{b_jl}y)
 ]
 converges almost everywhere as kto∞.
 
 Moreover, for each cin[1,2] and ε>0 there exists a standard
-averaging profile varphi with
+averaging profile phi with
 [
- norm{varphi-c^{-1}mathbf 1_bf1_{(0,c]}}_{L^1(R)}<ε.
+ norm{phi-c^{-1}mathbf 1_bf1_{(0,c]}}_{L^1(R)}<ε.
 ]
 
 
@@ -327,31 +327,31 @@ By scaling we may assume |g_j|le1.  Put S=R^g and
 [
  F_l=prod_{j=1}^3 g_j(S^{b_j l}y).
 ]
-Then |F_l|le 1.  Let N_k=lfloor c2^krfloor for cin[1,2].
+Then |F_l|le 1.  Let N_k=floor( c2^k) for cin[1,2].
 
 The smooth-to-ordinary step.
-For varphi a profile, we have
+For phi a profile, we have
 [
- V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)F_l.
+ V_k^phi(y)=sum_{linZ}2^{-k}phi(l/2^k)F_l.
 ]
 For h_c=c^{-1}mathbf 1_bf1_{(0,c]} set
 [
  W_k(y)=sum_{linZ}2^{-k}h_c(l/2^k)F_l.
 ]
 Then W_k=(1/N_k)sum_{l=1}^{N_k}F_l+O(1/(c2^k)) because h_c is the
-normalized indicator of (0,c], using lfloor c2^krfloor=N_k and the
+normalized indicator of (0,c], using floor( c2^k)=N_k and the
 error on the last c2^k-N_k terms being at most 1/(c2^k).  Also
 [
- |V_k^varphi(y)-W_k(y)|le 2^{-k}sum_{linZ}|h_c(l/2^k)-varphi(l/2^k)|.
+ |V_k^phi(y)-W_k(y)|le 2^{-k}sum_{linZ}|h_c(l/2^k)-phi(l/2^k)|.
 ]
-By Riemann sums the right side tends to norm{h_c-varphi}_1 uniformly in
-y.  Now pick a sequence of profiles varphi^{(m)} with
-norm{h_c-varphi^{(m)}}_1to0.  On a common conull set (intersection
+By Riemann sums the right side tends to norm{h_c-phi}_1 uniformly in
+y.  Now pick a sequence of profiles phi^{(m)} with
+norm{h_c-phi^{(m)}}_1to0.  On a common conull set (intersection
 of the conull sets supplied by Proposition~prop:input-smooth at each
-m) all V_k^{varphi^{(m)}}(y) converge.  For mto∞ the above
+m) all V_k^{phi^{(m)}}(y) converge.  For mto∞ the above
 display and the triangle inequality show that W_k(y) is Cauchy.  Since
 W_k and the ordinary average differ by O(2^{-k}), the ordinary averages
-along N_k=lfloor c2^krfloor form a Cauchy sequence for mu-almost
+along N_k=floor( c2^k) form a Cauchy sequence for mu-almost
 every y.  This holds for every c in a countable dense subset of
 [1,2].
 
@@ -362,7 +362,7 @@ For any bounded sequence |H_l|le1 and positive integers Mle N,
 ]
 For 2^kle N<2^{k+1} pick a finite rational grid cin[1,2] with mesh
 less than some eta, and compare N to the sequence values at
-lfloor c2^krfloor.  The error is at most O(eta).  Letting etato0
+floor( c2^k).  The error is at most O(eta).  Letting etato0
 on a countable union of scales preserves almost everywhere convergence.
 Therefore the average along every complex of lengths converges almost
 everywhere.
@@ -537,7 +537,7 @@ Set
 
  C=Xtimes[0,1), m_C=mu ⊗ dd u,
  
- S_C^t(x,u)=(T^{lfloor u+trfloor}x,{u+t}),
+ S_C^t(x,u)=(T^{floor( u+t)}x,{u+t}),
 
 and write xi(x,u)=x, υ(x,u)=u, viewing υ also as a
 coordinate on R/Z.  Then (C,m_C,S_C) is a measure-preserving flow.
@@ -551,7 +551,7 @@ phase coordinate, with
  N_a(x,u)in{c2^k:cin F, kge a},
  N_a(x,u)=2^a (xnotin E_*),
 
-and such that B_{lfloor N_a(x,u)rfloor}(x)>delta on E_*.  Moreover
+and such that B_{floor( N_a(x,u))}(x)>delta on E_*.  Moreover
 there is a continuous nonnegative function Psi on (R/Z)^I with
 
  liminf_{ato∞}int_Cfrac1{N_a(z)}
@@ -569,20 +569,20 @@ For a bounded sequence |H_l|le1 and positive integers Lle M,
  le(2|M-L|)/(max(M,L)).
 ]
 Pick a finite grid F⊆[1,2] containing 1, so fine that for every
-integer L there exist cin F and k=lfloorlog_2Lrfloor with
+integer L there exist cin F and k=floor(log_2L) with
 |L-c2^k|<eta L; choose eta so small that the length-change bound
 stays below delta.  Then for every xin E_* and every L at which
 B_L(x)>2delta, the corresponding dyadic average
-B_{lfloor c2^krfloor}(x) exceeds delta.
+B_{floor( c2^k)}(x) exceeds delta.
 
 The measurable choice.
 Fix an enumeration of the countable set
 {(c,k):cin F,,kinN}.  For ainN and xin E_*, let (c,k) be
 the first pair in this enumeration with kge a and
-B_{lfloor c2^krfloor}(x)>delta.  On E_* we set N_a(x,u)=c2^k;
+B_{floor( c2^k)}(x)>delta.  On E_* we set N_a(x,u)=c2^k;
 on X∖ E_* we set N_a=2^a.  As a map Cto[2^a,∞) this is
 measurable and independent of u.  By construction
-B_{lfloor N_a(x,u)rfloor}(x)>delta on E_*.
+B_{floor( N_a(x,u))}(x)>delta on E_*.
 
 The phase witness.
 Choose continuous nonnegative circle functions alpha,beta, not
@@ -605,12 +605,12 @@ condition, we obtain, uniformly in z and Lge2,
  prod_{iin J_0}f_i(xi(S_C^{it}z))dd t
  notag
  &hspace{18mm}
- =alpha(u)(int_0^1beta(s)dd s)B_{lfloor Lrfloor}(x)
+ =alpha(u)(int_0^1beta(s)dd s)B_{floor( L)}(x)
  +O(L^{-1}).
  
 
 The error covers the incomplete last unit interval, the change of index
-summation from 0,...,lfloor Lrfloor-1 to 1,...,lfloor Lrfloor,
+summation from 0,...,floor( L)-1 to 1,...,floor( L),
 and the normalisation.
 
 Integrating.
@@ -619,7 +619,7 @@ On E_*, with L=N_a(z), the integral of the main term in
 [
  deltamu(E_*)(int_0^1alpha(u)dd u)(int_0^1beta(s)dd s)>0,
 ]
-since B_{lfloor N_a(z)rfloor}>delta on E_*.  On X∖ E_*, the
+since B_{floor( N_a(z))}>delta on E_*.  On X∖ E_*, the
 function N_aequiv 2^a is independent of u, and B_{2^a}to0 in L^2(mu)
 by Lemma~lem:norm-vanishes; hence the contribution of this region
 tends to zero.  The uniform error is O(2^{-a}).  Taking liminf over
@@ -628,7 +628,7 @@ a completes the proof.
 
 ## Selected laws on every flow over C
 
-Fix the selectors and a free ultrafilter mathcal UcalU on N.
+Fix the selectors and a free ultrafilter U on N.
 
 [Flow over C; extensions]
 A flow over C is a standard probability space (Y,m_Y) with a
@@ -647,11 +647,11 @@ Every flow Y over C has a unique probability lambda_Y on Y^I
 satisfying, for all bounded measurable G_i:YtoR,
 
  int_{Y^I}prod_{iin I}G_i(y_i)ddlambda_Y
- =lim_{atomathcal U}calUint_Yfrac1{N_a(pi_Yy)}
+ =lim_{atoU}{N_a(pi_Yy)}
  int_0^{N_a(pi_Yy)}prod_{iin I}G_i(S_Y^{it}y)dd tdd m_Y(y).
 
 Its marginals are m_Y, and lambda_Y is invariant under
-(y_i)_{iin I}mapsto(S_Y^{it}y_i)_{iin I} for every tinR.  If
+(y_i)_{iin I}|->(S_Y^{it}y_i)_{iin I} for every tinR.  If
 rho:Y'to Y is an extension over C, then (rho^I)_*lambda_{Y'}=lambda_Y.
 
 Countable inverse limits exist in this category.  The selected law at the
@@ -663,13 +663,13 @@ Proof.
 Construction at each a.
 For each a, the expression before the ultralimit in (eq:selected-law)
 defines a probability measure lambda_{Y,a} on Y^I, since N_acircpi_Y
-is measurable, ymapsto S_Y^{it}y is jointly Borel, and the integrand is
+is measurable, y|-> S_Y^{it}y is jointly Borel, and the integrand is
 bounded.
 
 Marginals.
 We show lambda_{Y,a}to m_Y on marginals as ato∞ along every
 bounded G:YtoR.  By Birkhoff applied to the auxiliary function
-ymapstoint_0^1G(S_Y^{is}y)dd s and to S_Y^i, the continuous-time
+y|->int_0^1G(S_Y^{is}y)dd s and to S_Y^i, the continuous-time
 averages (1/L)int_0^LG(S_Y^{it}y)dd t converge almost everywhere as
 Lto∞, and their a.e. limit barbar GG satisfies
 int_Ybarbar GGdd m_Y=int_YGdd m_Y since every deterministic average has
@@ -678,7 +678,7 @@ int_Y(1/N_a)int_0^{N_a}G(S_Y^{it}y)dd tdd m_Y(y)toint_YGdd m_Y.
 
 Ultralimit.
 Embed Y into a compact metrizable K (as a Borel subset with m_Y
-extended by zero).  Along the ultrafilter mathcal U,calU,
+extended by zero).  Along the ultrafilter U,
 lambda_{Y,a} against continuous functions on K^I converges to a
 positive normalised functional, hence a probability K^I.  Its marginals
 are m_Y, so it is concentrated on Y^I.  Denote it lambda_Y.
@@ -804,19 +804,19 @@ one chosen slot as the original system.
 
 For bin I, define
 
- mathsf M_b(Y)=(Y^I,sfM_b(Y)=(Y^I,lambda_Y,
-   S_{mathsf M_b(Y)}sfM_b(Y)^t(y_i)_{iin I}=(S_Y^{it/b}y_i)_{iin I}).
+ M_b(Y)=(Y^I,lambda_Y,
+   S_{M_b(Y)}{iin I}=(S_Y^{it/b}y_i)_{iin I}).
 
 This is a standard probability-preserving flow over C via
-(y_i)_{iin I}mapstopi_Y(y_b); the projection onto slot b is an
+(y_i)_{iin I}|->pi_Y(y_b); the projection onto slot b is an
 extension onto Y.  The array P belongs to the maximal measure-distal
-factor of mathsf M_b(Y)sfM_b(Y) for time one.
+factor of M_b(Y) for time one.
 
 
 Proof. 
 Invariance follows from Proposition~prop:selected-law after rescaling
 the time parameter by 1/b.  The factor map is pi_Ycirc p_b.  Slot b
-has speed 1 and marginal m_Y, giving (y_i)_imapsto y_b as a factor
+has speed 1 and marginal m_Y, giving (y_i)_i|-> y_b as a factor
 onto Y.  In slot i, the W_Y-factor is distal for time i/b, by
 Assertion 2 (equivalence for rational powers); the law of P is an
 invariant joining of these rational-time distal systems, hence distal.
@@ -929,15 +929,15 @@ supersystem, gives the stronger conditioning on W_{Y'} for the extension
 Y'.
 
 
-## The multiplicative construction mathsf M_bsfM_b
+## The multiplicative construction M_b
 
 Recall from Lemma~lem:normalized-joining that for bin I,
-mathsf M_b(Y)sfM_b(Y) has state space Y^I, measure lambda_Y, flow
-(y_i)_{iin I}mapsto(S_Y^{it/b}y_i)_{iin I}, and factor y_bto Y.
+M_b(Y) has state space Y^I, measure lambda_Y, flow
+(y_i)_{iin I}|->(S_Y^{it/b}y_i)_{iin I}, and factor y_bto Y.
 
 [Channel closure]
 
-Let Y be a channel and bin I.  Define data on Z=mathsf M_b(Y)sfM_b(Y) by:
+Let Y be a channel and bin I.  Define data on Z=M_b(Y) by:
  the original factor is x_0(y_b)=xicircpi_Y(y_b);
  the outputs are all D_gamma(y_i), iin I, each in the old group
   of gamma;
@@ -959,9 +959,9 @@ conditional on P=(omega_Y(y_i))_{iin I}.
 
 Proof. 
 Apply Proposition~prop:channel-closure and
-Lemma~lem:channel-conditioning to mathsf M_b(Y).sfM_b(Y).  By
+Lemma~lem:channel-conditioning to M_b(Y).  By
 Lemma~lem:normalized-joining, the old parameter array P belongs to
-the maximal distal factor of mathsf M_b(Y).sfM_b(Y).  Conditioning the
+the maximal distal factor of M_b(Y).  Conditioning the
 constant product law (eq:channel-laws) down to P proves the
 assertion.
 
@@ -1008,7 +1008,7 @@ argument is T^{hbl+a_0}x_0, with the same finite-offset property.
 Equivariance and Fubini justify these formulas inside the time integrals.
 
 On an event where the selected length is N=c2^k (cin F, kge a), put
-m=lfloor c2^k/hrfloor.  Replacing the normalized time integral by the
+m=floor( c2^k/h).  Replacing the normalized time integral by the
 average over m complete intervals, followed by integration against
 ds/h, has error at most 2h2^{-a}.  For all sufficiently large a,
 mge1 on each admissible event.
@@ -1083,7 +1083,7 @@ determined by V,x_0, which is independent of any proper set of groups by
 (eq:channel-laws).  The full three-group law may change.  Its
 density rho relative to the product law satisfies
 
- 0le prhole1,qquadint prho=p,qquadnorm{prho}_2lesqrt p.
+ 0le prhole1, int prho=p, norm{prho}_2lesqrt p.
 
 The bound 0le prhole1 is the absolute-continuity statement; the second
 identity is the normalisation; the third is Cauchy--Schwarz.  These hold
@@ -1092,7 +1092,7 @@ for almost every V, and the length events are countable.
 The rank bound.
 The rank parameter must meet two requirements: the packing bound must be
 o(m), and the reciprocal bounds 1/d_m must be summable over dyadic
-length scales.  For m=lfloor c2^k/hrfloor set
+length scales.  For m=floor( c2^k/h) set
 
  d_m=lceil(1+log m)^{3/2}rceil, z_l=u_l ⊗  v_lquad(0le l<m).
 
@@ -1142,7 +1142,7 @@ event probability, is at most √(p/d_m).  Multiplication by the fixed
 phase test of absolute value at most 1 does not increase either bound.
 
 Summing over lengths.
-Write m(k,c)=lfloor c2^k/hrfloor.  For fixed h this is comparable to
+Write m(k,c)=floor( c2^k/h).  For fixed h this is comparable to
 2^k uniformly in cin F.  Since sum_{k,c}p_{k,c}(V)=1,
 Cauchy--Schwarz gives