aa r X i v : . [ m a t h . D S ] S e p A NEW DYNAMICAL PROOF OF THE SHMERKIN–WU THEOREM T IM A USTINDepartment of MathematicsUniversity of California, Los AngelesLos Angeles, CA 90095-1555, USAA
BSTRACT . Let a ă b be coprime positive integers, both at least , and let A, B be closedsubsets of r , s that are forward invariant under multiplication by a , b respectively. Let C “ A ˆ B . An old conjecture of Furstenberg asserted that any planar line L not parallelto either axis must intersect C in Hausdorff dimension at most max t dim C, u ´ . Tworecent works by Shmerkin and Wu have given two different proofs of this conjecture. Thisnote provides a third proof. Like Wu’s, it stays close to the ergodic theoretic machinerythat Furstenberg introduced to study such questions, but it uses less substantial backgroundfrom ergodic theory. The same method is also used to re-prove a recent result of Yu aboutcertain sequences of sums.
1. I
NTRODUCTION
Let T : “ R { Z . For any integer a ě , we write S a for either of two transformations: ‚ x ÞÑ ax on T ; ‚ x ÞÑ t ax u on r , q , where t¨u denotes fractional part.The obvious bijection r , q ÝÑ T identifies these two transformations, justifying ouruse of a single notation. We sometimes leave the correct choice of domain to the context.Similarly, for any u P T , we write R u for the rotation of T by u , or for the correspondingtransformation of r , q .Let a ă b be coprime integers, both at least . Let A, B be closed subsets of r , s thatare forward invariant under S a , S b respectively, and let C “ A ˆ B . Two recent papers,by Shmerkin [Shm19] and Wu [Wu19], independently prove the following conjecture ofFurstenberg [Fur70]. Theorem A (Shmerkin–Wu theorem) . If L is any line not parallel to either coordinateaxis, then dim p C X L q ď max t dim C, u ´ . Here and in the rest of the paper, dim denotes Hausdorff dimension. In a few places wealso need upper box dimension, which is denoted by dim B .Theorem A strengthens the classical slicing theorem of Marstrand, which provides thesame inequality of dimensions for almost all lines when C is any planar set. Furstenberg’soriginal formulation of Theorem A concerns intersections of affine images of the sets A and B , and for this reason it is often called Furstenberg’s intersection conjecture. It isequivalent to the above simply by writing the intersection C X L in coordinates, as healready explains in his paper. Mathematics Subject Classification.
Primary: 11K55, 37A45; Secondary: 28A50, 28A80, 37C45.
Key words and phrases.
Entropy, Hausdorff dimension, multiplication-invariant sets, intersections of Cantorsets, Furstenberg intersection conjecture, Shannon–McMillan–Breiman theorem.
TIM AUSTIN
Both Shmerkin’s and Wu’s proofs of Theorem A use ergodic theory, but in very differentways. Shmerkin’s approach is quite quantitative, and sets much of Furstenberg’s machineryfrom [Fur70] aside. His main results also have several other applications besides TheoremA. Wu’s work stays closer to Furstenberg’s methods, but finds a way to use major additionalresults from abstract ergodic theory, particularly the unilaterial Sinai factor theorem. Inthe present note we offer a new proof of Theorem A. It takes as its starting point one ofFurstenberg’s original results (Theorem 4.1 below), and then uses different backgroundfrom ergodic theory than Wu’s.In the rest of this paper, Section 2 is devoted to some background results from geometricmeasure theory. Section 3 gives a new proof of a recent theorem of Yu [Yu20]. This is notneeded for the proof of Theorem A, but it offers a simple model setting to illustrate ourapproach. Finally, Section 4 explains the new proof of Theorem A.2. S
OME PRELIMINARIES ON H AUSDORFF MEASURE AND DIMENSION
Let p X, ρ q be a metric space. For d ě , ε ą , and A Ă X , we write m p ε q d A : “ inf ! ÿ n p diam F n q d : x F n y a covering of A by sets of diam ă ε ) . Then m ˚ d A : “ lim ε Ó m p ε q d A is the d -dimensional Hausdorff outer measure of A . Itsrestriction to the Borel sets defines a true measure, the d -dimensional Hausdorff measure m d . The properties of these measures and the resulting notion of Hausdorff dimension canbe found in standard treatments such as [Fal14].2.1. A generalized Marstrand theorem.
In the proofs below we use a generalization ofMarstrand’s classical slicing theorem. I believe this generalization is widely known, but Ido not know of a reference that contains the exact version we need, so I include a full proofhere. It is similar to [Hoc, Theorem 8.1], but the proof below is closer to more classicalversions: compare, for instance, [Fal14, Corollary 7.10].
Proposition 2.1.
Let X and Y be metric spaces and let µ be a finite Borel measure on Y .Equip X ˆ Y with the max-metric for definiteness. Assume there exists a ą such that µB p y, r q ď r a ´ o p q as r Ó for µ -a.e. y , where the rate implied by the notation o p q may depend on the point y .Let W Ă X ˆ Y , and let W y : “ t x : p x, y q P W u for each y P Y . Then dim W y ` a ď max t dim W, a u for µ -a.e. y. Proof. Step 1.
We first complete the proof under the stronger assumption that there exists r ą such that µB p y, r q ď r a whenever y P Y and ă r ď r . Let d : “ dim W . Wemay assume this is finite, for otherwise the result is trivial.Having done so, let d ą max t d, a u , let ε ă r , and let x F n y be a covering of W bysets of diameter less than ε such that ÿ n p diam F n q d ď ε. For each n , let G n : “ t y : p x, y q P F n for some x P X u . Then W y ˆ t y u Ă ď n : G n Q y F n HMERKIN-WU THEOREM 3 for each y P Y , and hence m p ε q d ´ a p W y q ď ÿ n G n p y q ¨ p diam F n q d ´ a . Now we integrate against dµ p y q . Since diam G n ď diam F n ď r for each n , our strength-ened assumption on µ gives ż m p ε q d ´ a p W y q dµ p y q ď ÿ n µG n ¨ p diam F n q d ´ a ď ÿ n p diam G n q a ¨ p diam F n q d ´ a ď ÿ n p diam F n q d ď ε. Here we use the upper Lebesgue integral ş because we do not know that m p ε q d ´ a p W y q is ameasurable function of y . Since ε ą was arbitrary, it follows that m d ´ a p W y q “ @ d ą max t d, a u and hence dim W y ď max t d, a u ´ a. for µ -a.e. y . Step 2.
Now we return to our original hypothesis on µ . Let a P p , a q , and let Y m : “ t y P Y : µB p y, r q ď r a @ r ă { m u for each m P N . Then µ p Ť m Y m q “ , so it suffices to prove the result when y P Y m for some m . But forthis purpose we may replace Y with Y m , W with W X p X ˆ Y m q , and µ with µ p ¨ X Y m q .This returns us to the stronger hypothesis of Step 1, except with a in place of a . So nowwe conclude from Step 1 that dim W y ` a ď max t dim W, a u for a µ -a.e. y . Now let a increase through a sequence of values to a . (cid:3) Adic versions of the mass distribution principle.
The mass distribution principleis one of the most basic and versatile sources of lower bounds on Hausdorff measure anddimension. Standard formulations can be found in [Fal14, Section 4.1] or [Hoc, Proposition4.2].For one- and two-dimensional Euclidean subsets, we need some versions of this prin-ciple with adic intervals taking the place of balls. Such variants are well-known, but weinclude the precise statements we need for completeness.Let a ě and n be integers. An a -adic interval of depth n is a real interval of theform r ka ´ n , p k ` q a ´ n q for some k P Z . For any x P R , let I n p x q denote the a -adicinterval of depth n that contains x . If we fix n ě and let x vary in r , q , then the sets I n p x q constitute a partition of r , q . These partitions become finer as n increases. Theycan serve as substitutes for centred intervals in the mass distribution principle: Lemma 2.2.
Let µ be a finite Borel measure on r , q , and let d ě . Then the followingare equivalent:a. µ p z ´ r, z ` r q ď r d ´ o p q as r Ó for µ -a.e. z ;b. µI n p z q ď a ´ dn ` o p n q as n ÝÑ 8 for µ -a.e. z .If these conditions hold, then any Borel set with positive µ -measure has Hausdorff dimen-sion at least d . (cid:3) TIM AUSTIN
The equivalence of (a) and (b) is a special case of [Hoc, Proposition 6.21], and the finalimplication is the mass distribution principle.Now let b ě be another integer, and let J n p y q denote the b -adic interval of depth n that contains a point y P R . Let u : “ log a { log b . Then, for each n P N and p x, y q P R ,let(1) D n p x, y q : “ I n p x q ˆ J m p y q , where m “ t un u . This choice of m is the largest integer for which b ´ m ě a ´ n . As a result, the rectangle D n p x, y q is always close to being square: its height is at least its width, but no more than b times its width.If we fix n and let p x, y q vary in r , q , then the sets D n p x, y q constitute a partition of r , q . These partitions become finer as n increases. They participate in a two-dimensionalvariant of the mass distribution principle: Lemma 2.3.
Let µ be a finite Borel measure on r , q , and let d ě . Then the followingare equivalent:a. µB p z, r q ď r d ´ o p q as r Ó for µ -a.e. z ;b. µD n p z q ď a ´ dn ` o p n q as n ÝÑ 8 for µ -a.e. z .If these conditions hold, then any Borel set with positive µ -measure has Hausdorff dimenionat least d . (cid:3) This time the equivalence of (a) and (b) is not quite a special case of [Hoc, Proposition6.21], because that reference applies to true b -adic squares. However, the proof requiresonly that the rectangles D n p x, y q have uniformly bounded aspect ratios, so it carries overwithout change. 3. A THEOREM OF Y U Before approaching Theorem A, we give a new proof of a recent theorem of Yu [Yu20]:
Theorem 3.1.
Let u P T be irrational, let a ě be an integer, and let v P T be arbitrary.Then the closure of the set t nu ` a n v : n P N u has Hausdorff dimension . Yu proves this result by an adaption of Wu’s chief innovation in [Wu19], which is acertain application of the unilateral Sinai factor theorem referred to as the ‘Berrnoulli de-composition method’. Our next topic is a new, shorter proof of Theorem 3.1 which avoidsthis machinery. We include it here as a warmup to the coming proof of Theorem A.As remarked in his paper, Yu’s method really proves the following. Let σ : T ˆ T ÝÑ T be the map p u, v q ÞÑ u ` v . Theorem 3.2. If C Ă T ˆ T is nonempty, closed and forward invariant under R u ˆ S a ,then dim σ r C s “ . This implies Theorem 3.1 by letting C be the forward orbit closure of the point p , v q under R u ˆ S a .Let π i for i “ , denote the first and second coordinate projections T ˆ T ÝÑ T .Since C is closed and forward invariant under R u ˆ S a , its image π r C s is closed andforward invariant under S a . Let h be the topological entropy of the topological dynamicalsystem p π r C s , S a q . A classical calculation of Furstenberg [Fur67, Proposition III.1] gives dim π r C s “ dim B π r C s “ h log a . We denote this value by d in the remainder of this section. HMERKIN-WU THEOREM 5
Lemma 3.3.
The set C carries a Borel probability measure µ which is invariant and er-godic under R u ˆ S a and such that p T , π ˚ µ, S a q has Kolmogorov–Sinai entropy equal to h .Proof. Since S a is expansive, the topological dynamical system p π r C s , S a q has a measureof maximal entropy ν , whose Kolmogorov–Sinai entropy therefore equals h . Replacing ν with one of its ergodic components if necessary, we may assume it is ergodic. Let µ be any p R u ˆ S a q -invariant and ergodic lift of this measure to C . (cid:3) We fix the measures µ and ν “ π ˚ µ for the rest of this section. The image of µ under π is an invariant measure for R u , so must equal Lebesgue measure m by unique ergodicity.In addition, let µ “ ż T δ t ˆ ν t dt be the disintegration of µ over π .Since p T , m, R u q has entropy zero, h is also equal to ‚ the Kolmogorov–Sinai entropy of p T , µ, R α ˆ S a q , and ‚ the relative Kolmogorov–Sinai entropy of p T , µ, R α ˆ S a q over π .Let Σ a : “ t , , . . . , a ´ u , and let α : T ÝÑ Σ α be the map such that α p x q isthe first digit in the a -ary expansion of x . The sequence of observables α n : “ α ˝ S na , n P N , generates the whole Borel sigma-algebra of T , so by the Kolmogorov–Sinaitheorem the resulting partitions of T witness the full Kolmogorov–Sinai entropy of thesystem p π r C s , ν, S a q .By ergodicity and the relative Shannon–McMillan–Breiman theorem [EW, Theorem3.2], these entropy values imply that(2) ν t I n p x q “ e ´ hn ` o p n q “ p a ´ n q d ` o p q as n ÝÑ 8 for m -a.e. t and then for ν t -a.e. x . This asymptotic is the starting point of ourdimension estimates, via Lemma 2.2. Proof of Theorem 3.2.
Let C t “ t y P T : p t, y q P C u , and observe that “ µC “ ż T ν t C t dt, and so ν t C t “ for a.e. t . Therefore dim C t ě d for a.e. t , by (2) and Lemma 2.2.Since π r C s is nonempty, closed and forward invariant under R u , it must be the wholeof T . This implies that dim C ě dim T “ , and so Proposition 2.1 gives dim C t ` ď dim C for m -a.e. t. Combining these properties, we find that a.e. t is a witness to the inequality(3) d ` ď dim C. Next, consider the map p σ, π q : T ˆ T ÝÑ T ˆ T : p x, y q ÞÑ p x ` y, y q . This is bi-Lipschitz, so it preserves the dimensions of subsets, and we obtain(4) dim C “ dim ` p σ, π qr C s ˘ ď dim p σ r C s ˆ π r C sq ď dim σ r C s ` d, where the last inequality is a standard bound for the Hausdorff dimension of a prod-uct [Fal14, Product formula 7.3], combined with the fact that dim B π r C s “ d . Concaten-ing the inequalities (3) and (4), the terms involving d cancel to leave dim σ r C s ě . (cid:3) TIM AUSTIN
Remark 1.
The set σ r C s may miss a large portion of the circle T , even though it has fulldimension. Indeed, if x u n y is any sequence in T , then v P T may be chosen one decimaldigit at a time so that the number u n ` n v lies within the interval r , { s modulo forevery n . Then the closure of the set t u n ` n v : n P N u is entirely contained in thissmall interval. 4. P ROOF OF T HEOREM
AWe now return to the setting of the Introduction.4.1.
Furstenberg’s auxiliary transformation.
Let u : “ log a { log b as before. This isirrational since a and b are coprime. Ergodic theory enters the proof of Theorem A becauseof the following specific transformation. On the space X “ C ˆ T , we define T p x, y, t q : “ " p S a x, S b y, R ´ u t q if ď t ă u p S a x, y, R ´ u t q if u ď t ă . This is a skew product over the irrational circle rotation R ´ u . Upon iteration, it yields T n p x, y, t q “ ` S na x, S m p t,n q b y, R n p ´ u q t ˘ , where m p t, n q is the number among the points t, t ` p ´ u q , t ` p ´ u q , . . . , t ` p n ´ qp ´ u q that lie in the interval r , u q mod . In the sequel we need the estimate(5) | m p t, n q ´ un | ď @ t, n, which is [Fur70, Lemma 7]. These constructs are discussed further in [Wu19, Section 5].Given z P R and t P R , let L z,t be the line in R that has slope b t and passes through z . The next theorem is a corollary of [Fur70, Theorem 9], which is the technical heart ofthat paper. It is a kind of analog of Lemma 3.3 for the purposes of this section, but it ismuch more substantial than that lemma. Theorem 4.1.
If any planar line, not parallel to either coordinate axis, intersects C indimension at least γ , then there is a probability measure µ on X which is invariant andergodic for T and such that dim p C X L z,t q ě γ for µ -a.e. p z, t q . In the remainder of the paper we use the invariant measure µ promised by this theoremto prove that γ is at most max t dim C, u ´ .Let µ be the projection of µ to C . The image of µ under the final coordinate projectionto T is invariant under the irrational rotation R ´ u , so it must be Lebesgue measure m byunique ergodicity. Let(6) µ “ ż T µ t ˆ δ t dt be the disintegration of µ over that projection to T .Let h be the Kolmogorov–Sinai entropy of the system p X, µ, T q , and let d : “ h { log a .Then h is also the relative entropy of p X, µ, T q over the coordinate projection to p T , m, R ´ u q ,since the latter system has entropy zero. HMERKIN-WU THEOREM 7
Consequences of the Shannon–McMillan–Breiman theorem.
Let Σ a : “ t , , . . . , a ´ u ,and define Σ b similarly. Let α : X ÝÑ Σ a be the map such that α p x, y, t q is the first digitin the a -ary expension of x , and let β be the analogous map which reports the first digit inthe b -ary expansion of y . For each integer n ě , let α n : “ α ˝ T n and β n : “ β ˝ T n , andlet α r n s p z, t q : “ p α p z, t q , . . . , α n p z, t qq and β r n s p z, t q : “ p β p z, t q , . . . , β n p z, t qq . In this notation, α r n ´ s p x, y, t q lists the first n digits in the a -ary expansion of x . Simi-larly, β r n ´ s p x, y, t q lists the first m p t, n q digits in the b -ary expansion of y , but with rep-etitions so that the output is a sequence of length n . See the discussion preceding [Fur70,Lemma 7].For each n ě , the level sets of the combined map p α r n s , β r n s q constitute a partitionof X . We write P n p z, t q for the cell of this partition that contains p z, t q : that is, P n p z, t q : “ p z , t q : α r n s p z , t q “ α r n s p z, t q and β r n s p z , t q “ β r n s p z, t q ( . These partitions P n generate the system p X, µ, T q relative to its zero-entropy factor p T , m, R ´ α q ,so they witness the full Kolmogorov–Sinai entropy of the system p X, µ, T q .The next lemma compares the cells P n with other, simpler subsets of X . To formulate it,we endow R with the max-metric, and let B p z, r q denote the open ball of radius r around z P R in this metric. Lemma 4.2.
There is a fixed positive integer c such that D n ` c p x, y q ˆ t t u Ă P n p x, y, t q Ă B pp x, y q , ba ´ n q ˆ T for all p x, y q P C , t P T and n P N , where D n is the notation from (1) .Proof. Choose c large enough that uc ě , and suppose that p x , y q P D n ` c p x, y q . Then x and x agree in the first n ` c digits of their a -adic expansions, and y and y agree in thefirst m digits of their b -adic expansions, where m “ r u p n ` c qs “ r un ` uc s ě r un s ` ě un ` , where the first inequality follows from our choice of c . In view of (5), this implies that m ě m p t, n q , and so α r n s and β r n s both agree on the two inputs p x, y, t q and p x , y , t q .This proves the first inclusion.For the second, observe from (5) that if p x , y , t q P P n p x, y, t q then x , x must liewithin distance a ´ n and y , y must lie within distance b ´ un ` “ ba ´ n . Hence p x , y q must lie within distance ba ´ n of p x, y q . This proves the second inclusion. (cid:3) We are now ready to derive estimates on the local behaviour of various measures on C :first the projection µ , and then the disintegrand µ t in (6) for typical values of t . Proposition 4.3.
The measure µ has the property that (7) µB p z, r q ě r d ` o p q as r Ó for µ -a.e. z , where the rate implied by the notation o p q may depend on the point z .Proof. Equivalently, we must show that µ ` B p z, r q ˆ T ˘ ě r d ` o p q for µ -a.e. p z, t q . TIM AUSTIN
By the second inclusion of Lemma 4.2, this measure is bounded below by µP n p z, t q ,where n is the smallest integer for which ba ´ n ď r . By the Shannon–McMillan–Breimantheorem, we have µP n p z, t q “ e ´ hn ´ o p n q “ a ´ h log a n ´ o p n q “ r d ` o p q as n ÝÑ 8 for µ -a.e. p z, t q . (cid:3) Corollary 1.
There is a Borel subset C of C such that µC “ and dim C ď d .Proof. By Proposition 4.3, there is a Borel set C such that (7) holds for every z P C .Now a standard covering argument shows that m s C “ for all s ą d , so dim C ď d :see, for instance, in [Fal14, Proposition 4.9(b)] or [Hoc, Proposition 6.24]. (cid:3) Now we turn to the disintegrands in (6).
Proposition 4.4.
For m -a.e. t , the measure µ t has the property that µ t B p z, r q ď r d ´ o p q as r Ó for µ t -a.e. z , where the rate implied by the notation o p q may depend on the pair p z, t q .Proof. By Lemma 2.3, it is equivalent to show that µ t D n p z q ď a ´ dn ` o p n q as n ÝÑ 8 for µ t -a.e. z . This fact has a similar proof to Proposition 4.3, except thatnow we must use the relative Shannon–McMillan–Breiman theorem [EW, Theorem 3.2].Indeed, the first inclusion of Lemma 4.2 gives µ t D n p z q “ p µ t ˆ δ t qp D n p z q ˆ t t uq ď p µ t ˆ δ t qp P n ´ c p x, y, t qq for all sufficiently large n . By the relative Shannon–McMillan–Breiman theorem, thisright-hand side behaves as e ´ h p n ´ c q` o p n q “ e ´ hn ` o p n q “ a ´ dn ` o p n q as n ÝÑ 8 for m -a.e. t and then for µ t -a.e. p x, y q . (cid:3) Completion of the proof.
Completed proof of the Theorem A. Step 1.
From the product set C ˆ C , we constructthe following new sets: ‚ For each t P r , q , let D t : “ p z, w q P C ˆ C : w P L t,z zt z u ( . ‚ Let D : “ p z, w q P C ˆ C : w P L z,t zt z u for some t P r , q ( “ ď t Pr , q D t . In prose, this is the set of pairs p z, w q of distinct points in C such that (i) z lies inthe subset C and (ii) the segment from z to w has positive slope lying in r , b q . ‚ Finally, let D : “ p z, w, t q P C ˆ C ˆ r , q : w P L z,t zt z u ( . This is similar to D , except it records explicitly the slope between z and w . HMERKIN-WU THEOREM 9
The next few steps of the proof estimate and relate the dimensions of these sets.
Step 2.
By Furstenberg’s calculation for multiplication-invariant sets in [Fur67, Propo-sition III.1], both A and B are exact dimensional: that is, each has equal Hausdorff andupper and lower box dimensions. Using this fact, standard formulae for dimensions forproducts [Fal14, Product formulae 7.2, 7.3] give (i) that C is also exact dimensional, andthen (ii) these estimates:(8) dim D ď dim p C ˆ C q “ dim C ` dim C ď d ` dim C. Step 3.
The slope between z and w is a Lipschitz function of the pair p z, w q when werestrict to any set of the form tp z, w q P D : | z ´ w | ě { n u , n P N . Therefore D is acountable union of Lipschitz images of subsets of D , and so dim D ď dim D . The reverseinequality here is immediate, so in fact dim D “ dim D . Step 4.
Applying Proposition 2.1 to the set D and the projection p z, w, t q ÞÑ t , wehave(9) dim D t ` ď max t dim D, u “ max t dim D, u for m -a.e. t . Combining this fact with our previous results, it follows that m -a.e. t satis-fies (9) and also:i. dim p C X L z,t q ě γ for µ t -a.e. z (by Theorem 4.1);ii. µ t B p z, r q ď r d ´ o p q as r Ó for µ t -a.e. z (by Proposition 4.4);iii. µ t C “ , where C is the set provided by Corollary 1 (in view of the disintegra-tion (6) and the fact that µC “ ).Fix such a value of t for the rest of the proof. Step 5.
Consider the restricted coordinate projection π : D t ÝÑ C : p z, w q ÞÑ z. By property (iii) above, the measure µ t is supported by the target C of this map. Under thismap, the pre-image π ´ t z u is precisely the set t z u ˆ pp C X L z,t qzt z uq , whose dimensionis equal to dim p C X L z,t q . Therefore property (ii), Lemma 2.3, and another appeal toProposition 2.1 give(10) dim p C X L z,t q ` d ď max t dim D t , d u for µ t -a.e. z . Finally, the left-hand side here is at least γ ` d for µ t -a.e. z , by property (i).If dim D t ď d , then (10) implies at once that γ “ , which complies with Theorem A.On the other hand, if dim D t ą d , then the right-hand side in (10) equals dim D t . In thiscase we concatenate inequalities (8), (9) and (10) to obtain γ ` d ` ď max t d ` dim C, u . If this maximum is equal to , then γ “ d “ , and otherwise we can cancel d to concludethat γ ď dim C ´ . (cid:3) A CKNOWLEDGEMENTS
I am grateful to Adam Lott for reading an early version of this manuscript and pointingout several corrections. R EFERENCES[EW] Manfred Einsiedler and Thomas Ward. Entropy in ergodic theory and homogeneous dynamics. Bookdraft, available online at https://tbward0.wixsite.com/books/entropy .[Fal14] Kenneth Falconer.
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