Directional Poincare inequalities along mixing flows
aa r X i v : . [ m a t h . C A ] M a r DIRECTIONAL POINCAR´E INEQUALITIESALONG MIXING FLOWS
STEFAN STEINERBERGER
Abstract.
We provide a refinement of the Poincar´e inequality on the torus T d : there exists aset B ⊂ T d of directions such that for every α ∈ B there is a c α > k∇ f k d − L ( T d ) k h∇ f, α i k L ( T d ) ≥ c α k f k dL ( T d ) for all f ∈ H ( T d ) with mean 0.The derivative h∇ f, α i does not detect any oscillation in directions orthogonal to α , however,for certain α the geodesic flow in direction α is sufficiently mixing to compensate for that defect.On the two-dimensional torus T the inequality holds for α = (1 , √
2) but fails for α = (1 , e ).Similar results should hold at a great level of generality on very general domains. Introduction and main result
Introduction.
The classical Poincar´e inequality on the torus T d states k∇ f k L ( T d ) ≥ k f k L ( T d ) for functions f ∈ H ( T d ) with vanishing mean. A natural interpretation is that a function withsmall derivatives cannot substantially deviate from its mean on a set of large measure. The purposeof this paper is to derive a substantial improvement; we first state the main result. Theorem 1 (Directional Poincare inequality) . There exists a set
B ⊂ T d such that for every α ∈ B there is a c α > so that k∇ f k d − L ( T d ) k h∇ f, α i k L ( T d ) ≥ c α k f k dL ( T d ) for all f ∈ H ( T d ) with mean 0. If d ≥ , then B is uncountable but Lebesgue-null. The exponents are optimal. The proof is simple and based on elementary properties of Fourierseries – we believe it to be of great interest to understand under which conditions comparableinequalities exist on a general Riemannian manifold (
M, g ) equipped with a suitable vector field.
Figure 1.
A well-mixing flow transports (dashed) every point relatively quicklyto a neighborhood of every other point.
Mathematics Subject Classification. [email protected] . On the torus, the inequality has strong ties with number theory and can be easily derived at thecost of invoking highly nontrivial results (Schmidt’s result on badly approximable numbers, theKhintchine theorem). One remarkable feature is that the inequality holds for a set of Lebesguemeasure 0 which shows it to be very delicate (however, as is explained below, slightly weakerstatements seem to be very robust). A natural interpretation of the inequality seems to be thefollowing: given two nearby points x, y ∈ T d for which f ( x ) ≫ f ( y ), the classical Poincar´einequality will detect a large gradient between them. The term | h∇ f, α i | might not detect thelarge gradient but following the ergodic vector field will relatively quickly lead to a neighborhoodof y . A priori being in a neighborhood might not imply much because there could be still localoscillations on the scale of the neighborhood, however, since we also invoke a power of k∇ f k L ( T d ) ,this controls the measure of the set on which local oscillations have a strong effect. This heuristicsuggests strongly that similar inequalities should hold at a much greater level of generality. Wediscuss and prove some natural variants in the last section.1.2. Open problems.
It would be of great interest to understand to which extent such inequali-ties can be true in a more general setup. It is not even clear to us whether comparable inequalitieshold in L p ( T d ). Generally, for suitable vector fields Y on suitable Riemannian manifolds ( M, g )it seems natural to ask whether there exists an inequality of the type k∇ f k − δL p ( M ) k h∇ f, Y i f k δL p ( M ) ≥ c k f k L p ( M ) for some δ > f ∈ W ,p ( M ) with mean 0. The parameter δ can be expected to be relatedto the mixing properties of the flow – it is difficult to predict what the generic behavior on a fixedmanifold might be (say, for a smooth perturbation of the flat metric on the torus). On T we canrephrase the Khintchine theorem [11] as a statement about generic behavior for the flat metric. Theorem (Khintchine, equivalent) . For every δ < / , the set of α ∈ T for which there exists a c α > such that ∀ f ∈ H ( T ) Z T f ( x ) dx = 0 = ⇒ k∇ f k − δL ( T ) k h∇ f, α i f k δL ( T ) ≥ c α k f k L ( T ) has full measure. This suggests δ < / T , the manifolds on whichthe inequality holds with δ = 1 / S d equipped with a nontrivial vector field: the hairy ball theoremdictates that any smooth vector field vanishes for even d and this will necessitate a change of scalingin the inequality since a function f could be concentrated around the point in which the vectorfield vanishes. Furtermore, while not every nonvanishing vector field on S has to have a closedorbit (i.e. Seifert’s conjecture is false), many of them do – this puts topological restrictions onwhat directional Poincar´e inequalities are possible (since one could set a function to be constantalong a periodic orbit and decaying extremely quickly away from it). However, there should bea variety of admissible inequalities on the flat infinite cylinder ( M, g ) = ( R × T d − , can) and thiscould be a natural starting point for future investigations.2. Proof of the Statements
Outline of the argument.
The proof of the classical Poincar´e inequality on the torus is aone-line argument if one expands in Fourier series and uses ˆ f (0) = R T d f = 0 since k∇ f k L ( T d ) = (2 π ) d X k ∈ Z dk =0 | k | | a k | ≥ (2 π ) d X k ∈ Z dk =0 | a k | = k f k L ( T d ) . The argument also highlights the underlying convexity of the quadratic form. Our proof will be adirect variation of that result and uses the observation that k h∇ f, α i k L ( T d ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)* X k ∈ Z d a k ke ik · x , α +(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( T d ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ Z d a k h k, α i e ik · x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( T d ) = (2 π ) d X k ∈ Z d | a k | | h k, α i | . This Fourier multiplier is not uniformly bounded away from 0 and will even vanish for certain k ∈ Z d if the entries of α are not linearly independent over Q . If the entries of α are linearlyindependent over Q , then the Fourier multiplier is always nonnegative but we have no quantitativecontrol on its decay (see below for an example). However, if we restrict the support of the Fouriercoefficients to be within (cid:8) k ∈ Z d : | k | ≤ R (cid:9) , then trivially k h∇ f, α i k L ( T d ) ≥ inf k ∈ Z d | k |≤ R | h k, α i | k f k L ( T d ) . The term in the bracket clearly has great significance in the study of geometry of numbers andhas been studied for a long time. It suffices for us to apply the results and use the additional k∇ f k L ( T d ) expression to ensure that a fixed proportion of the L − mass is contained within asuitable ball of frequeny space on which to apply the argument.2.2. Number theoretical properties.
We now discuss subtleties of the inequality in greaterdetail: it is merely the classical Poincar´e inequality for d = 1. Letting d = 2 with α = (1 ,
0) yields k∇ f k L ( T ) k ∂ x f k L ( T ) ≥ c k f k L ( T ) which is obviously false,because f might be constant along the x − direction and vary along the y − direction. More gen-erally, the inequality fails for any α with entries linearly dependent over Q and the functionssin ( k x + k y ) for any k , k ∈ Z with h ( k , k ) , α i = 0 serve as counterexamples. The nextnatural example is α = ( √ , . Suppose f ∈ C ∞ ( T ) and (cid:13)(cid:13)(cid:13)D ∇ f, ( √ , E(cid:13)(cid:13)(cid:13) L ( T ) = 0 .f is constant along the flow of the vector field ( √ ,
1) but every orbit is dense and thus f ≡ Q , however, it is notenough to prove the inequality itself: it fails for (1 , e ) on T despite linear independence. A simpleconstruction for d = 2 shows that linear independence of the entries of α is not enough: let α = , ∞ X n =1 n ! ! ∼ (1 , . . . . )where the arising number, Liouville’s constant, is known to be irrational. If we set f N ( x, y ) = sin N ! N X n =1 x n ! − y !! , then k f N k L ( T ) = 2 π and k∇ f N k L ( T ) ≤ · N ! while kh∇ f N , α ik L ( T ) = √ π ∞ X n = N +1 N ! n ! ! ≪ − · N ! for N ≥ . An explicit example.
The inequalities are not only sharp with respect to exponents, theyare actually sharp on all frequency scales . This is in stark contrast to classical Poincar´e-typeinequality which tend to be sharp for one function (the ground state of the underlying physicalsystem): here, we can exclude all functions having Fourier support in the set { ξ : | ξ | ≤ N } forarbitrarily large N and still find functions for which the inequality is sharp (up to a constant).We explain this in greater detail for d = 2 with the admissible direction given by the golden ratio α = , √ ! ∈ B . Consider the sequence of functions given by f n ( x, y ) = sin ( F n +1 x − F n y ) , where F n is the n − th Fibonacci number. An explicit computation shows that k∇ f n k L ( T ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ x f n + 1 + √ ∂ y f n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( T ) = s F n +1 F n + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n +1 F n − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n k f n k L ( T ) A standard identity for Fibonacci numbers gives thatlim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n +1 F n − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n = 1 √ , which implies thatlim n →∞ k∇ f n k L ( T ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ x f n + 1 + √ ∂ y f n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( T ) k f n k − L ( T ) = s √
510 = | α |√ . Since these functions f n have their Fourier transform supported on 4 points in Z and since F n +1 /F n → (1 + √ / <
2, we can conclude that every dyadic annulus in Fourier space containsan example for which the inequality is sharp (up to a constant). Put differently, our inequalityis close to being attained on every frequency scale. This sequence of f n has the advantage ofsimultaneously showing that the following statement is sharp. Proposition.
Let d = 2 and α ∈ T be any vector for which k∇ f k L ( T ) k h∇ f, α i k L ( T ) ≥ c α k f k L ( T ) holds for all f ∈ H ( T ) with mean 0. Then the constant satisfies c α ≤ | α |√ . Up to certain transformation, the example above is essentially the only example for which theinequality is tight: normalizing α = (1 , β ), the example shows that the inequality is sharp for β = (1 + √ / β for which it is sharp (that can beexplicitely given). For all other numbers the upper bound could be improved to c α ≤ | α | / √ √ √ / Badly approximable systems of linear forms.
We now introduce the relevant resultsfrom number theory. Let L , . . . , L ℓ : Z d → R be defined as L ( x ) = α x + . . . α d x d = h α , x i . . .L ℓ ( x ) = α ℓ x + . . . α ℓd x d = h α ℓ , x i The relevant question is whether it is possible for all ℓ expressions to be very close to an integer.Using k·k : R → [0 , /
2] to denote the distance to the closest integer, the pigeonhole principleimplies the existence of infinitely many x ∈ Z d withmax( k L ( x ) k , . . . , k L ℓ ( x ) k ) ≤ (max( | x | , . . . , | x d | )) − dℓ . Dirichlet’s theorem cannot be improved in the sense that there actually exist badly approximablevectors α , . . . , α ℓ such that for some c > x ∈ Z d max( k L ( x ) k , . . . , k L ℓ ( x ) k ) ≥ c (max( | x | , . . . , | x d | )) − dℓ . The existence of such elements was first shown by Perron [15]. Khintchine [11] has shown that the ℓd − dimensional Lebesgue measure of such tuples ( α , . . . , α ℓ ) is 0 and Schmidt [18] has proventhat their Hausdorff dimension is ℓd .2.5. Proof of Theorem 1.
Proof.
We will prove the statement explicitely for the following set: for any ( d − − dimensionalbadly approximable vector α d − , consider the linear form L : Z d − → R given by L ( x ) := h α d − , x i and α = (1 , α d − ) . Recall that k·k : R → [0 , /
2] denotes the distance to the closest integer and is trivially 1-periodic.Let now k ∈ Z d with k = 0. If k vanishes on all but the first component, then | h α, k i | = | k | ≥ . If k does not vanish on all but the first component, then | h α, k i | = | k + L (( k , . . . , k d )) | ≥ || L (( k , . . . , k d )) || ≥ c max( | k | , . . . , | k d − | ) d − ≥ c | k | d − , where c > α d − is badly approximable. Let now f = X k ∈ Z d a k e ik · x ∈ H ( T d )and note that a = 0 because f has mean value 0. We have k h∇ f, α i k L ( T d ) = (2 π ) d X k ∈ Z d | a k | | h k, α i | ≥ c (2 π ) d X k ∈ Z d | a k | | k | d − . It is easy to see that X | k |≥ k∇ f k L T d ) k f k L T d ) | a k | ≤ k f k L ( T d ) k∇ f k L ( T d ) = X k ∈ Z d | k | | a k | ≥ X | k |≥ k∇ f k L T d ) k f k L T d ) | k | | a k | ≥ k∇ f k L ( T d ) k f k L ( T d ) X | k |≥ k∇ f k L T d )2 k f k L T d )2 | a k | ≥ k∇ f k L ( T d ) , which is absurd. Altogether, we now have k h∇ f, α i k L ( T d ) ≥ c (2 π ) d X k ∈ Z d | a k | | k | d − ≥ c (2 π ) d X k ∈ Z d | k |≤ k∇ f k L T d ) k f k L T d ) | a k | | k | d − ≥ c (2 π ) d d − k f k d − L ( T d ) k∇ f k d − L ( T d ) X | k |≤ k∇ f k L T d ) k f k L T d ) | a k | ≥ c (2 π ) d d − k f k d − L ( T d ) k∇ f k d − L ( T d ) k f k L ( T d ) . Rearranging gives the result. (cid:3)
We remark that a classical insight of Liouville allows to give a completely self-contained proof inthe most elementary case. If we pick α = ( √ , c > | h k, α i | = | k √ k | ≥ c | k | for all k ∈ Z . However, this follows at once with c = 1 / ≤ | k − k | = | ( √ k − k )( √ k + k ) | ≤ | k || ( √ k + k ) | . A similar argument works for any (1 , α ) with α algebraic over Q (Liouville’s theorem). Moregenerally, a classical characterization of badly approximable numbers in one dimension as thosenumbers with a bounded continued fraction expansion implies that our proof works for α = (1 , β ) ∈ R if β has a bounded continued fraction expansion. A theorem of Lagrange (see e.g. [12]) impliesthat this is always the case if β is a quadratic irrational. Moreover, this characterization is sharpon T : if β has an unbounded continued fraction expansion, then the inequality is not true for(1 , β ) and the sequence f n ( x ) = e πik n · x with k n =(numerator, -denominator) of rational approximations of β coming from the continuedfraction expansion will serve as a counterexample. A very interesting special case is Euler’scontinued fraction formula for e (see, e.g. [10]), which implies that e has an unbounded continuedfraction expansion and that the inequality with α = (1 , e ) fails on T .2.6. Proof of the Proposition.
Proof.
The direction α has to have both entries different from 0. We use a classical result ofHurwitz [7] which guarantees the existence of infinitely many k ∈ Z with (cid:12)(cid:12)(cid:12)(cid:12) α α − k k (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ k . For any such ( k , k ), this can be rewritten as | α k − α k | ≤ | α |√ | k | . We now consider f ( x ) = e πik · x . Simple computation yields k∇ f k L ( T ) k h∇ f, α i k L ( T ) ≤ √ | α || k || k | k f k L ( T ) . However, as | k | → ∞ , we have that k /k → α /α and thus | α || k || k | = | α | s k k + 1 −→ | α | s α α + 1 = | α | . (cid:3) The constant in the result of Hurwitz is sharp for the golden ratio α = α /α = (1 + √ / φ .Moreover, it is known (see e.g. [2]) that for every α ∈ R \ Q which is not of the form aφ + bcφ + d a, b, c, d ∈ Z | ad − bc | = 1 , the constant √ √
8. Our example showing the sharpness of the Propositionusing Fibonacci numbers was therefore, in some sense, best possible.2.7.
Fractional derivatives.
As is obvious from the proof, fine properties of the derivative didnot play a prominent role, indeed, the proof really only requires an understanding of how fast theinduced Fourier multiplier grows. This allows for various immediate generalizations. We introducepseudodifferential operators P ( D ) on H s ( T d ) via P ( D ) X k ∈ Z d a k e ik · x := X k ∈ Z d a k P ( k ) e ik · x the same proof can immediately be applied as long as | P ( k ) | → ∞ if | k | → ∞ . One example on T would be that for α ∈ B and all s > k∇ s f k s L ( T ) k h∇ f, α i i k L ( T ) ≥ c α k f k s L ( T ) which is again sharp by the same reasoning as above. The following variant was proposed byRaphy Coifman: if we define D s X k ∈ Z d a k e ik · x := X k ∈ Z d a k k | k | s − e ik · x , then one can always observe substantial fluctuations in the ( d − − th derivative along the flow (cid:13)(cid:13)(cid:10) D d f, α (cid:11)(cid:13)(cid:13) L ( T d ) ≥ c α k f k L ( T d ) . Several ergodic directions.
We can also derive a statement for more than one ergodicdirection. The same heuristics as above still apply: the main difference is that incorporating acontrol in more than one ergodic direction poses additional restrictions and requires less globalcontrol in the sense that a proportionately smaller power of k∇ f k L ( T d ) is necessary. Theorem 2.
Let ≤ ℓ ≤ d − . Then there exists a set B ℓ ∈ ( T d ) ℓ such that for every ( α , α , . . . , α ℓ ) ∈ B ℓ there is a c α > with k∇ f k d − L ( T d ) ℓ X i =1 k h∇ f, α i i k L ( T d ) ! ℓ ≥ c α k f k d − ℓL ( T d ) for all f ∈ H ( T d ) with mean 0.Proof. We consider ℓ ≤ d − β , β , . . . , β ℓ from T d − with ℓ associated linear forms L i : Z d − → R via L i = h β i , x i such that they form a system of badly approximable linear formsand set α = (1 , β ) . . .α ℓ = (1 , β ℓ )The same reasoning as before (distinguishing between k vanishing outside of the first componentor not) implies again for every single 1 ≤ i ≤ ℓ | h α i , k i | = | k + L i (( k , . . . , k d )) | ≥ || L i (( k , . . . , k d )) || from which we derive that whenever k is not concentrated on the first component ℓ X i =1 | h α i , k i | ≥ max( k L ( k ) k , . . . , k L ℓ ( k ) k ) ≥ c (max( | k | , . . . , | k d | )) − d − ℓ . If k is concentrated on the first component, we get a bound of ℓ , which is much larger. For f = X k ∈ Z dk =0 a k e ik · x a simple computation shows that ℓ X i =1 k h∇ f, α i i k L ( T d ) = (2 π ) d X k ∈ Z d | a k | ℓ X i =1 | h k, α i i | ! ≥ c X k ∈ Z d | a k | (max( | k | , . . . , | k d | )) − d − ℓ ≥ c X k ∈ Z d | a k | | k | − d − ℓ . The rest of the argument proceeds as before; finally, we recall that any two norms in finite-dimensional vector spaces are equivalent and thus, up to some absolute constants depending onlyon ℓ , ℓ X i =1 k h∇ f, α i i k L ( T d ) ! ℓ ∼ ℓ ℓ X i =1 k h∇ f, α i i k L ( T d ) ! ℓ and the result follows. (cid:3) The Hausdorff dimension.
As is obvious from the proof, we are not so much interest inthe distance to the lattice but care more about the distance to the origin. It seems that there isongoing research in that direction [4, 8, 9], which is concerned with establishing bounds on thedimension of the set max( | L ( x ) | , . . . , | L ℓ ( x ) | ) ≥ c (max( | x | , . . . , | x d | )) − dℓ +1 , where | · | is the absolute value on R . For any such system of linear forms given by α , . . . , α ℓ satisfying that inequality, we can improve Theorem 2 with the same proof to k∇ f k d − ℓL ( T d ) ℓ X i =1 k h∇ f, α i i k L ( T d ) ! ℓ ≥ c α k f k dL ( T d ) for all f ∈ H ( T d ) with mean 0. We also remark the following simple proposition (the essence ofwhich is contained in [8]). Proposition.
Let d ≥ . α ∈ R d is admissible in Theorem 1 if and only if there exists λ ∈ R such that α = ( λ, λβ ) and β is badly approximable linear form in R d − . Using the result of Schmidt [18], we see that the Hausdorff dimension of the set of badly approx-imable vectors in R d − is d − d . The argument is indirectlycontained in the earlier proofs. Variants.
This subsection is concerned with inequalities of the type k∇ f k − δL ( T ) k h∇ f, α i f k δL ( T ) ≥ c k f k L ( T ) for some 0 < δ ≤ /
2. The case δ = 1 / δ > / δ = 1 / Theorem (Khintchine) . For every δ < / , the set of α ∈ T for which there exists a c α > suchthat k∇ f k − δL ( T ) k h∇ f, α i f k δL ( T ) ≥ c k f k L ( T ) holds for all f ∈ L ( T ) with mean 0 has full measure. Another celebrated result in Diophantine approximation is the Thue-Siegel-Roth theorem statingthat for every irrational algebraic number α and every ε > (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ c α q ε for some c α >
0. This immediately implies, along the same lines as above, that for every vectorof the form α = (1 , β ) with β being an irrational algebraic number and every ε > k∇ f k / εL ( T ) k h∇ f, α i f k / − εL ( T ) ≥ c α,ε k f k L ( T ) . Recall that the inequality does probably not hold for α = (1 , π ) because π is probably not badlyapproximable. However, there are weaker positive results. A result of Salikhov [17] implies theexistence of a c > (cid:12)(cid:12)(cid:12)(cid:12) π − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq . Repeating again the same argument as above, we can use this to derive k∇ f k / L ( T ) k h∇ f, (1 , π ) i k / L ( T ) ≥ c k f k L ( T ) . Similarly, a result of Marcovecchio [13] shows k∇ f k / L ( T ) k h∇ f, (1 , log 2) i k / L ( T ) ≥ c k f k L ( T ) and similar results are available for other numbers (i.e. π , log 3 , ζ (3) , . . . ).3. Uniform Poincar´e inequalities via diffusion
There exists a closely related problem, where we can try to substitute a directional derivative by arandom process (that points roughly in the direction of its mean but may also behave differently).Let T = R \ Z be the one-dimensional torus T , let f ∈ L p ( T ) and consider the averaging operator A t : L p ( T ) → L p ( T ) given by ( A t f )( x ) = 12 t Z + t − t f ( x + y ) dy. An iterative application of A t corresponds to a time-discrete diffusion. These objects have strongand natural ties to Poincar´e inequalities (see, for example, the recent book by Bakry, Gentil andLedoux [1]). One natural question is whether k f − A t f k L p ( T ) can be small if f is not constant.Motivated by a problem of Schechtman, this was recently investigated by Nayar & Tkocz [14].They studied the problem in the general case of convolution with a T − valued random variable Y which for some ℓ ∈ N Y + Y + · · · + Y ℓ has a nontrivial absolutely continuous part. Theorem 3 (Nayar & Tkocz) . There exists a constant c > depending only on Y such that forall f ∈ L p ( T ) , ≤ p ≤ ∞ , with R T f = 0 k f ( x ) − E f ( x + tY ) k L p ( T ) ≥ ct k f k L p ( T ) . We believe the inequality to be of quite some interest; the example above corresponds to Y beinguniformly distributed on T but the setting is clearly more general than that. It is easy to see thatthe inequality is sharp: for a fixed C ( T ) − function f and Y being again the uniform distributionit follows from a Taylor expansion thatlim t → k f − A t f k L p t = 16 k f ′′ k L p . Using C ( T ) ֒ → L p ( T ), the statement immediately implies a Poincar´e inequality k f ′′ k L p ( T ) ≥ c k f k L p ( T ) with a constant c > p . There is a natural variant: note that for any f ∈ C the case E Y = 0 yields a first order contributionlim t → E ( f ( x + tY )) − f ( x ) t = ( E Y ) f ′ ( x ) . This suggests that the cases E ( Y ) = 0 and E ( Y ) = 0 behave differently: from the point of view ofdiffusion, the case E Y = 0 turns the averaging operator E f ( x + tY ) into an operator with drift,which allows for a faster exploration of the torus. We prove that this is indeed the case. Theorem 4.
Assume E Y = 0 . Then there exists a constant c > depending only on Y such thatfor all f ∈ L p ( T ) , ≤ p ≤ ∞ with R T f = 0 k f ( x ) − E f ( x + tY ) k L p ( T ) ≥ ct k f k L p ( T ) . It is easy to see that the statement cannot be improved: consider Y ∼ U ([0 , / , /
2] and let f = χ [0 , / − χ [1 / , . Then k f k L p ( T ) = 1 and for all 1 ≤ p < ∞k f ( x ) − E f ( x + tY ) k L p ( T ) ∼ t. By taking the limit t →
0, we obtain k f ′ k L p ( T ) ≥ c k f k L p ( T ) with a constant c > p . We will first prove the statements and thencomment on the relation with the material presented above.3.1. The proof of Nayar & Tkocz.
We start by giving a summary of the proof given in [14]since our argument for Theorem 1 is based on that approach. We write the averaging operator f ( x ) → E f ( x + tY ) as a convolution f → f ∗ µ. If k f − f ∗ µ k L p ( T ) is small, then f ∗ µ ∼ f and repeating the argument suggests that k f ∗ µ − f ∗ µ ∗ µ k L p ( T ) should be small as well.Since µ comes from a probability measure, Young’s convolution inequalities implies that k f ∗ µ − f ∗ µ ∗ µ k L p ( T ) = k ( f − f ∗ µ ) ∗ µ k L p ( T ) ≤ k f − f ∗ µ k L p ( T ) . The next ingredient is the triangle inequality allowing us to compare k f − f ∗ µ k L p ( T ) ≥ n (cid:0) k f − f ∗ µ k L p ( T ) + k f ∗ µ − f ∗ µ k L p ( T ) + · · · + k f ∗ µ n − − f ∗ µ n k L p ( T ) (cid:1) ≥ n k f − f ∗ µ n k L p ( T ) , where µ k := µ ∗ µ ∗ · · · ∗ µ | {z } k timesis the k − fold convolution of µ with itself. The final ingredient of the argument is the following: if n is sufficiently large, then the n − fold convolution µ n is very well-behaved: its absolutely continuouspart satisfies a uniform lower bound. However, if µ n ≥ c in a pointwise sense and f has mean 0,then [14, Lemma 2] k f ∗ µ k L p ( T ) ≤ (1 − c ) k f k L p ( T )1 and therefore k f − f ∗ µ k L p ( T ) ≥ n k f − f ∗ µ n k L p ( T ) ≥ n (cid:0) k f k L p ( T ) − k f ∗ µ n k L p ( T ) (cid:1) ≥ cn k f k L p ( T ) . Figure 2.
Flow of an expanding heat kernel along a geodesic.It remains to find the right value of n for which µ n ≥ c . It is easy to see that this generally scaleslike n ∼ /t for sufficiently nice µ and it remains to show that µ n is sufficiently nice for n largeenough assuming that µ ℓ has an absolutely continuous part for some ℓ . Summarizing, the heatkernel flattens and will eventually cover the entire torus; this requires roughly 1 /t convolutionswhich follows from the usual scaling that the variance of µ k scales as √ k . It is not surprising thata drift should allow for this exploration of the torus to happen at a different time scale.3.2. Random variables with drift: proof of Theorem 4.
Proof.
The argument of Nayar & Tkocz used the contraction property to establish k f − f ∗ µ k k L p ( T ) ≤ k k f − f ∗ µ k L p ( T ) . We will work with an averaged version of that statement: summing from k = 1 , . . . , n , we get1 n n X k =1 k f − f ∗ µ k k L p ( T ) ≤ n k f − f ∗ µ k L p ( T ) . Another application of the triangle inequality implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f − f ∗ n n X k =1 µ k !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( T ) ≤ n n X k =1 k f − f ∗ µ k k L p ( T ) ≤ n k f − f ∗ µ k L p ( T ) . The conclusion of the argument proceeds in the same way (though, obviously by a differentmethod): if we can conclude that1 n n X k =1 µ k ≥ c pointwise, then k f − f ∗ µ k L p ( T ) ≥ cn k f k L p ( T d ) . It remains to show that the statement is asymptotically true for n ∼ t − to conclude the result.We show that a result from renewal theory can now be applied relatively easily. Define Y R tobe Y on [ − / , / ⊂ R (with corresponding measure µ R ) and assume furthermore w.l.o.g. that E Y R >
0. We first note that if some convolution Y ℓ = Y + Y + · · · + Y ℓ on the torus has an absolutely continuous component, then so will Y ℓ R on R : this follows immedi-ately from the Radon-Nikodym theorem and the fact that the singular measure is supported ona set of measure 0 (moreover, Y l R is supported on [ − ℓ/ , ℓ/ µ R (1) has a finite m − th moment(2) µ ℓ R has a nontrivial continuous component for some ℓ ∈ N (3) and if furthermore E µ R > then the measure P ∞ k =1 µ k can be written as ∞ X k =1 µ k = ν + ν , where ν is absolutely continuous and has a density p ( x ) satisfying p ( x ) = ( o ( | x | − m ) as x → −∞ ( E Y R ) − + o ( | x | − m ) as x → ∞ and ν is a finite measure such that all integer moments exist. x Figure 3.
Heat kernels with drift.The result even says slightly more and allows to conclude exponential decay of ν as well as p ( x ) − / ( E Y R ) from exponential decay of µ R (which is something that is the case in our settingsince µ R is compactly supported but we will not use this fact). We now localize that statement(so as to avoid summing up to infinity). Note that µ k R has a large proportion of its mass centeredin an interval of length ∼ √ k centered around k ( E µ R ). This easily implies that, for n sufficientlylarge, P nk =1 µ k has an absolutely continuous part with a density p ( x ) satisfying p ( x ) ≥
12 1 E Y R for all x ≤ x ≤ n E Y R , where x is a universal number depending only on Y (essentially the threshold beyond which theStone asymptotic becomes effective). We remark that much sharper asymptotics could be derivedand one could prove the same result for x ≤ x ≤ ( n − C √ n ) E Y R for some fixed C > tY R is very simple because the entire setup scales nicelyunder dilations. If we define µ t as the probability measure of tY R . Then µ t has an absolutelycontinuous component p t ( x ) satisfying p t ( x ) ≥
12 1 t E Y R for all tx ≤ x ≤ n t E Y R . Let now n = 8 / ( t E Y R ) + 2 x / ( t E Y R ). We can conclude that p t ( x ) ≥
12 1 t E Y R on an interval of length at least 2 on R . Since n ∼ /t , this now implies that1 n n X k =1 µ kt, R has an absolutely continuous component with density ≥ c for some c > R . By projecting down on the torus, we have that1 n n X k =1 µ kt has an absolutely continuous component with density ≥ c and this implies the desired result by the reasoning described above. (cid:3) Concluding Remarks.
Let us try to understand the arguments above in a more generalcontext; the natural framework seems to be that of a compact Lie group and random variablesfor which the central limit theorem is valid. We are interested in the largest possible function h : R + → R + such that there exists a constant c > Y with the property thatfor all f ∈ L p , 1 ≤ p ≤ ∞ satisfying R f = 0 k f ( x ) − E f ( x + tY ) k L p ≥ h ( t ) k f k L p . The argument of Nayar & Tkocz implies that if E Y = 0 ∈ T d the best answer is given by h ( t ) ∼ t and this follows from the fact that the characteristic time for the heat kernel to effectively coverthe entire torus is of scale t − . If E Y = 0, then the situation becomes more interesting. Figure 4.
Location of the mass of the first few convolutions µ ℓ .Using standard approximations, we may assume that µ kt ( µ t again being the probability measureof tY ) has most of its mass around t E Y in a radius of scale ∼ √ t . This may be best understood asa moving and expanding bubble. The problem is to understand the number of bubbles N requiredto have 1 N N X k =1 µ k ≥ c. On T simple volumetric considerations imply that N & t − . However, it is also not difficult tosee that N . t − is false, even for badly approximable E Y : this would imply h ( t ) & t , which inturn would imply a Poincar´e type inequality kh∇ f, α ik L p & k f k L p , which we know to fail even for badly approximable directions. Another interesting example is thesphere, where one cannot improve on h ( t ) & t because every geodesic flow is periodic. It seemsinteresting to understand how the precise quantitative behavior in t depend on approximationproperties of E Y and, more generally, how such problems behave in different geometries. Acknowledgement.
I am grateful to Raphy Coifman for various discussions and to him, YvesMeyer and Jacques Peyri`ere for their encouragement.
References [1] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators. Grundlehren derMathematischen Wissenschaften, 348. Springer, 2014.[2] J. W. S. Cassels, An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Math-ematical Physics 45. Cambridge University Press.[3] T.W. Cusick and M. E. Flahive, The Markov and Lagrange Spectra. Providence, RI: Amer. Math. Soc., 1989.[4] H. Dickinson, The Hausdorff dimension of systems of simultaneously small linear forms. Mathematika 40 (1993),no. 2, 367–374.[5] P.G.L. Dirichlet, Verallgemeinerung eines Satzes aus der Lehre von den Kettenbruechen etc. S.B. preuss. Akad.W. (1842), 93–95. [6] D. Hensley, Continued fractions. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006[7] A. Hurwitz, Ueber die angenaeherte Darstellung der Irrationalzahlen durch rationale Brueche. MathematischeAnnalen 39 (2): 279–284.[8] M. Hussain, A note on badly approximable linear forms. Bull. Aust. Math. Soc. 83 (2011), no. 2, 262–266.[9] M. Hussain and S. Kristensen, Badly approximable systems of linear forms in absolute value. Unif. Distrib.Theory 8 (2013), no. 1, 7–15.[10] M. Iosifescu and C. Kraaikamp, Metrical theory of continued fractions. Mathematics and its Applications, 547.Kluwer Academic Publishers, Dordrecht, 2002.[11] A. Khintchine, Zur metrischen Theorie der diophantischen Approximationen. Mathematische Zeitschrift 24(1926), 706–714.[12] A. Khinchine, Continued fractions. The University of Chicago Press, Chicago, Ill. London 1964[13] R. Marcovecchio, The Rhin-Viola method for log 2. Acta Arith. 139 (2009), no. 2, 147-184.[14] P. Nayar and T. Tkocz, A note on certain convolution operators, Geometric Aspects of Functional AnalysisLecture Notes in Mathematics Volume 2116, 2014, pp 405–412.[15] O. Perron, ¨Uber diophantische Approximationen, Mathematische Annalen 84 (1921), 77–84.[16] Y. Prohorov (1952): On a local limit theorem for densities. Doklady Akad. Nauk SSSR 83, pp. 797–800.[17] V. Kh. Salikhov, On the irrationality measure of π . Uspekhi Mat. Nauk 63 (2008), no. 3(381), 163–164;translation in Russian Math. Surveys 63 (2008), no. 3, 570–572 .[18] W. Schmidt, Badly approximable systems of linear forms. J. Number Theory 1 1969 139–154.[19] C. Stone, On absolutely continuous components and renewal theory. Ann. Math. Statist. 37 1966 271–275. Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA
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