Characterization of Minkowski measurability in terms of surface area
aa r X i v : . [ m a t h . C A ] D ec CHARACTERIZATION OF MINKOWSKI MEASURABILITY INTERMS OF SURFACE AREA
JAN RATAJ AND STEFFEN WINTER
Abstract.
The r -parallel set to a set A in Euclidean space consists of allpoints with distance at most r from A . Recently, the asymptotic behaviour ofvolume and surface area of the parallel sets as r tends to 0 has been studiedand some general results regarding their relations have been established. Herewe complete the picture regarding the resulting notions of Minkowski contentand S-content. In particular, we show that a set is Minkowski measurable ifand only if it is S-measurable, i.e. if and only if its S-content is positive andfinite, and that positivity and finiteness of the lower and upper Minkowskicontents imply the same for the S-contents and vice versa. The results areformulated in the more general setting of Kneser functions. Furthermore,the relations between Minkowski and S-contents are studied for more generalgauge functions. The results are applied to simplify the proof of the ModifiedWeyl-Berry conjecture in dimension one. Introduction
Let A be a bounded subset of R d and r >
0. Denote by d A the (Euclidean)distance function of the set A , and by A r := { z ∈ R d : d A ( z ) ≤ r } its r -parallel set (or r -parallel neighbourhood). For t ≥
0, denote by H t the t -dimensional Hausdorff measure. Let V A ( r ) = H d ( A r ) be the volume of the r -parallelset.Stach´o [12] showed (using the results of Kneser [5]) that the derivative ( V A ) ′ ( r )of V A ( r ) exists for all r > r > V A ) ′− ( r ) ≥ ( V A ) ′ + ( r ). Combining someresults of Stach´o [12] and Hug, Last and Weil [4], it was shown in [11] using arectifiability argument that, for all r > V ′ A ( r ) = H d − ( ∂A r )holds.In [11], also the limiting behaviour of V A ( r ) and H d − ( ∂A r ) as r → A ⊂ R d and some close relations were establishedbetween the resulting notions of Minkowski content and S-content. Date : October 15, 2018.2000
Mathematics Subject Classification.
Key words and phrases. parallel set, surface area, Minkowski content, Minkowski dimension,gauge function, Kneser function.Acknowledgement: The authors were supported by a cooperation grant of the Czech and theGerman science foundation, GACR project no. P201/10/J039 and DFG project no. WE 1613/2-1.
Recall that the s -dimensional lower and upper Minkowski content of a compactset A ⊂ R d are defined by M s ( A ) := lim inf r → V A ( r ) κ d − s r d − s and M s ( A ) := lim sup r → V A ( r ) κ d − s r d − s , where κ t := π t/ / Γ(1 + t ). (If t is an integer, then κ t is the volume of a unit t -ball). If M s ( A ) = M s ( A ), then the common value M s ( A ) is refered to as the s -dimensional Minkowski content of A . If the Minkowski content M s ( A ) exists and ispositive and finite, then the set A is called ( s -dimensional) Minkowski measurable .We denote bydim M A := inf { t ≥ M s ( A ) = 0 } = sup { t ≥ M s ( A ) = ∞} and dim M A = inf { t ≥ M s ( A ) = 0 } = sup { t ≥ M s ( A ) = ∞} the lower and upper Minkowski dimension of A . Minkowski measurability playsan important role for instance in connection with the Weyl-Berry conjecture, seeSection 5.In analogy with the Minkowski content, the upper and lower S-content (or surfacearea based content ) of A was introduced in [11], for 0 ≤ s < d , by S s ( A ) := lim inf r → H d − ( ∂A r )( d − s ) κ d − s r d − − s and S s ( A ) := lim sup r → H d − ( ∂A r )( d − s ) κ d − s r d − − s , respectively. If both numbers coincide, the common value is denoted by S s ( A )and called ( s -dimensional) S-content of A . It is convenient to set S d ( A ) := 0for completeness, which is justified by the fact that lim r → r H d − ( ∂A r ) = 0 forarbitrary bounded sets A ⊂ R d , cf. [5]. If S s ( A ) exists and is positive and finite,then we call the set A ( s -dimensional) S-measurable . The numbersdim S A := sup { ≤ t ≤ d : S t ( A ) = ∞} = inf { ≤ t ≤ d : S t ( A ) = 0 } and dim S A := sup { ≤ t ≤ d : S t ( A ) = ∞} = inf { ≤ t ≤ d : S t ( A ) = 0 } are the lower and upper surface area based dimension or S-dimension of A , respec-tively. Obviously, dim S A ≤ dim S A , and if equality holds, the common value willbe regarded as the surface area based dimension (or S-dimension ) of A and denotedby dim S A .In view of equation (1.1), it is apparent that Minkowski contents and S-contentsof a set A must be closely related. In [11], some precise results regarding thisrelation have been obtained, which are summarized as follows. Theorem 1.1. [11, Corollaries 3.2, 3.4, 3.6 and Proposition 3.7]
Let A ⊂ R d be bounded and s ∈ [0 , d ] . Then d − sd S s ( A ) ≤ M s ( A ) ≤ S s ( A ) , (1.2) where for s = d the left inequality is trivial and the right inequality holds only incase V A (0) = 0 . As a consequence, dim M A = dim S A ,
HARACTERIZATION OF MINKOWSKI MEASURABILITY 3 whenever V A (0) = 0 . Furthermore, S s ( A ) ≤ M s ( A ) ≤ c d,s h S s d − d ( A ) i dd − , (1.3) where for the right hand inequality we assume d > and where the constant c d,s just depends on the dimensions s and d . As a consequence, dim S A ≤ dim M A ≤ dd − S A. (1.4)Note that there is a fundamental difference between upper and lower contents.While the upper contents differ at most by a positive constant implying in particularthe equivalence of the upper dimensions, the lower Minkowski content is in generalonly bounded from above by an S-content of some different dimension. This allowsdifferent lower dimensions. It was shown in [15], that there exist indeed sets forwhich lower Minkowski dimension and lower S-dimension are different. Moreover,the constants given in (1.4) were shown to be optimal.In this note, we complete the picture concerning the relations between Minkowskicontents and S-contents. We show that the existence of the Minkowski content (asa positive and finite number) is equivalent to the existence of the correspondingS-content, and that both numbers coincide in this case. In particular, this allowsto characterize Minkowski measurability in terms of S-measurability. Moreover,while the positivity of lower s -dimensional Minkowski contents is in general notenough to conclude the positivity of the corresponding S-content, we show that theassumption of both positivity and finiteness of the upper and lower s -dimensionalMinkowski contents is sufficient for the corresponding (upper and lower) S-contentsto be positive and finite as well. Hence two sided bounds for Minkowski contentsimply two sided bounds for the S-contents and vice versa, see Section 2.In Section 3, we study also contents with more general gauge functions. Moti-vated by an open question regarding the asymptotics of area and boundary lengthof the parallel sets of the Brownian path in R , we study generalized Minkowski con-tents, where the dimensions s (corresponding to the gauge functions h ( r ) = r d − s )are replaced by more general gauge functions h : (0 , ∞ ) → (0 , ∞ ) (e.g. h ( r ) = | log r | − as in the case of the Brownian path in the plane, cf. [10]). GeneralizedMinkowski contents are known in the literature, see e.g. [2, 3, 13, 16]. We introducethe corresponding generalization of the S-contents and extend the relations betweenMinkowski and S-contents to the generalized counterparts. In particular, we showfor a large class of gauge functions that the existence of the generalized Minkowskicontent M ( h ; A ) is equivalent to the existence of the corresponding generalizedS-content S ( h ′ ; A ), where h ′ is the derivative of h provided h is differentiable. Al-though our results cover a large class of gauge functions, unfortunately, they donot cover the case of the Brownian path. Our methods which are based on theKneser property do not apply in this particular case. In Section 4, some exam-ples of Kneser functions are constructed which indicate that the expected relationbetween the generalized contents may actually fail. At least they show that theunderlying result for Kneser functions is not valid in this case.Finally, in Section 5, we study subsets of R and demonstrate how the resultsin this note can be used to simplify some essential parts of the proof derived byLapidus and Pomerance in [8] of the Modified Weyl-Berry conjecture in dimensionone. JAN RATAJ AND STEFFEN WINTER Two sided bounds
In [11], we considered the relation between either the two upper contents orthe two lower contents, i.e. we tried to establish one-sided bounds, in which wesucceeded in the case of the upper contents, but which turned out to be impossiblefor the lower contents, see Theorem 1.1 above. While the lower S-content is alwaysa lower bound for the lower Minkowski content, cf. (1.3), it is impossible to boundthe lower S-content from below by the lower Minkowski content, if nothing is knownabout the upper Minkowski content. Even the lower S-dimension can be strictlysmaller than the lower Minkowski dimension. Now we consider upper and lowercontents together. Assuming both lower and upper Minkowski content (of thesame dimension D ) to be positive and finite, one can conclude the same for theS-contents. We will derive this from a statement on Kneser functions formulated inProposition 2.1 below. Recall that a function f : (0 , ∞ ) → (0 , ∞ ) is called Kneserfunction of order d ≥
1, if for all 0 < a ≤ b < ∞ and λ ≥ f ( λb ) − f ( λa ) ≤ λ d ( f ( b ) − f ( a )) . Stach´o observed that for a Kneser function f of order d ≥
1, the function f ′ + ( r ) r − d is monotone decreasing (cf. [12, Theorem 1]). It is not difficult to seethat the same is true for the function f ′− ( r ) r − d . That is, for 0 < t ≤ r ,(2.1) f ′− ( t ) t d − ≥ f ′− ( r ) r d − and f ′ + ( t ) t d − ≥ f ′ + ( r ) r d − . Note that a certain converse to Proposition 2.1 was formulated in [11, Proposi-tion 3.1], see also Proposition 3.1 below.
Proposition 2.1.
Let f be a Kneser function of order d ≥ . If (2.2) 0 < lim inf r → f ( r ) r s ≤ lim sup r → f ( r ) r s < ∞ for some s ∈ (0 , ∞ ) , then < lim inf r → f ′ + ( r ) r s − ≤ lim sup r → f ′− ( r ) r s − < ∞ . Proof.
It follows from the assumption that there exist 0 < m < M < ∞ and r > r ∈ (0 , r ), mr s ≤ f ( r ) ≤ M r s . (2.3)Choose a > ma s − M >
0. By (2.1), we have for any r > f ( ar ) − f ( r ) = Z arr f ′ + ( t ) dt ≤ Z arr f ′ + ( r ) (cid:18) tr (cid:19) d − dt = f ′ + ( r ) r a d − d . Moreover, we conclude from (2.3) that, for ar ≤ r , f ( ar ) − f ( r ) ≥ ( ma s − M ) r s . Combining the last two inequalities, we obtain f ′ + ( r ) r s − ≥ da d − ma s − M ) > r ∈ (0 , r a ), and thus lim inf r → f ′ + ( r ) r s − > HARACTERIZATION OF MINKOWSKI MEASURABILITY 5
A similar, slightly simpler argument works for the finiteness of the correspondinglimes superior. Choosing some 0 < b <
1, we have on the one hand f ( r ) − f ( br ) ≤ f ( r ) ≤ M r s , by (2.3), and on the other hand(2.4) f ( r ) − f ( br ) = Z rbr f ′− ( t ) dt ≥ f ′− ( r ) r − b d d , by (2.1). Combining both inequalities, we get f ′− ( r ) r s − ≤ Md − b d , which implies thefiniteness of the limsup.Note that the first part of the assumption (that the limes inferior for f is positive)is not needed for this conclusion. Since b can be chosen arbitrarily close to 0 and M arbitrarily close to lim sup r → f ( r ) r s , we obtain the relationlim sup r → f ′− ( r ) r s − ≤ d lim sup r → f ( r ) r s . (2.5) (cid:3) We point out that the statement on the limes superior can also be derived from[11, Lemma 3.5]. The latter is formulated for the volume function V A but extendsto arbitrary Kneser functions (cf. also [11, p.10, first Remark]). However, theargument given here is simpler. The constant obtained in (2.5) is the same. Theorem 2.2.
Let A be a bounded subset of R d and D ∈ [0 , d ) . Then < M D ( A ) ≤ M D ( A ) < ∞ (2.6) if and only if < S D ( A ) ≤ S D ( A ) < ∞ . (2.7) In this case, one has in particular dim M A = dim S A = D .Proof. Assume first that (2.6) holds. Let f ( r ) := V A ( r ) and recall that f is aKneser function of order d for each bounded set A in R d . Set s := d − D . Theassumptions imply that s > < lim inf r → ( V A ) ′ + ( r ) r d − D − ≤ lim sup r → ( V A ) ′− ( r ) r d − D − < ∞ . Now note that S A ( r ) ≥ ( V A ) ′ + ( r ) for each r >
0, which implies the positivity of S D ( A ), and that S A ( r ) ≤ ( V A ) ′− ( r ) for each r >
0, which allows to conclude thefiniteness of S D ( A ). This proves the implication (2.6) ⇒ (2.7).The reverse implication follows directly from Theorem 1.1, more precisely fromthe second inequality in (1.2) and the first inequality in (1.3). (cid:3) Refining the argument in the proof of Proposition 2.1, we will now establish,that the existence of the Minkowski content M D ( A ) of a bounded set A ⊂ R d of Minkowski dimension D ∈ [0 , d ) implies the existence of its S-content S D ( A ).This extends a result in [1] (where this was shown for D ≤ d −
1) and clarifies andsimplifies its proof.
Proposition 2.3.
Let f be a Kneser function of order d ≥ . If (2.8) lim r → f ( r ) r s = C JAN RATAJ AND STEFFEN WINTER for some constants s, C ∈ (0 , ∞ ) , then lim r → f ′ + ( r ) sr s − = lim r → f ′− ( r ) sr s − = C. Proof.
Let ε ∈ (0 , min { , C } ). Choose a > a s ≥ C + √ εC − √ ε . Since √ ε > ε and thus C + √ ε > C + ε and C − √ ε < C − ε , we have( C − ε ) a s − ( C + ε ) > . Repeating the argument of the first part of the proof of Proposition 2.1 with m := C − ε and M := C + ε , we infer that there exists some r = r ( ε ) such that f ′ + ( r ) r s − ≥ da d − C − ε ) a s − ( C + ε )) = dC a s − a d − − dε a s + 1 a d − > r ∈ (0 , r ). Hencelim inf r → f ′ + ( r ) r s − ≥ dC a s − a d − − dε a s + 1 a d − ε ∈ (0 , min { , C } ) and each a > a suchthat equality holds in (2.9) and let ε tend to 0. Then a = a ( ε ) converges to 1 andso does a t for each t ∈ R . Moreover, for the last term on the right hand side wehave lim sup ε → dε a s + 1 a d − ≤ lim ε → dε (cid:16) C + √ εC −√ ε (cid:17) ds − , which is easily seen by applying L’Hˆopital’s rule. Using again L’Hˆopital’s rule, weconclude that lim inf r → f ′ + ( r ) sr s − ≥ ds C lim a ց a s − a d − ds C lim a ց sa s − da d − = C .
An analogous argument shows thatlim sup r → f ′− ( r ) sr s − ≤ C. (Here one can choose b = b ( ε ) such that b s = ( C − √ ε ) / ( C + √ ε ) and derive theinequality lim sup r → f ′− ( r ) r s − ≤ dC − b s − b d + dε b s − b d from which the claim follows by letting again ε → f ′ + ≤ f ′− . (cid:3) Theorem 2.4.
Let A be a bounded subset of R d , D ∈ [0 , d ) and M ∈ (0 , ∞ ) . Then M D ( A ) = M, if and only if S D ( A ) = M. That is, the set A is Minkowski measurable (of order D ) if and only if it is S-measurable (of order D ) and in this case both ( D -dimensional) contents coincide. HARACTERIZATION OF MINKOWSKI MEASURABILITY 7
Proof.
Assume that M D ( A ) = M . Let f ( r ) := V A ( r ) and s := d − D and applyProposition 2.3. Then S D ( A ) = M follows from the fact that ( V A ) ′ + ≤ S A ≤ ( V A ) ′− .The reverse implication is a direct consequence of Theorem 1.1. (cid:3) General gauge functions
In the definition of generalized Minkowski contents the renormalization quotient r d − D is replaced by a more general gauge function h . This allows to characterizethe convergence behaviour of the parallel volume on a much finer scale, which isparticularly useful, when the ordinary Minkowski content (of the correct dimension)is zero or infinite. Generalized Minkowski contents have been studied for instancein [2, 3, 13, 16].For a continuous function h : (0 , ∞ ) → (0 , ∞ ), let M ( h ; A ) := lim inf r → V A ( r ) h ( r ) and M ( h ; A ) := lim sup r → V A ( r ) h ( r ) , be the lower and upper generalized Minkowski content with respect to h . Similarly,we define the lower and upper generalized S-content with respect to h by S ( h ; A ) := lim inf r → S A ( r ) h ( r ) and S ( h ; A ) := lim sup r → S A ( r ) h ( r ) . If the corresponding upper and lower limits coincide, their common value is de-noted by M ( h ; A ) and S ( h ; A ), respectively. In the sequel we will establish somerelations between generalized contents. It turns out that again derivatives play animportant role. We formulate our results first for general Kneser functions (andtheir derivatives) and specialize them afterwards to relations between volume andboundary surface area (or Minkowski and S-content).In [11], the following proposition was proved in order to establish bounds forthe Minkowski content in terms of the S-content. The result allows to immediatelyextend these bounds to the generalized contents. Proposition 3.1. [11, Proposition 3.1 and Remark on p.10]
Let f be a Kneserfunction and let h : (0 , ∞ ) → (0 , ∞ ) be a differentiable function with lim r → h ( r ) =0 . Assume that h ′ is nonzero on some right neighbourhood of .Let S := lim inf r → f ′ ( r ) /h ′ ( r ) and S := lim sup r → f ′ ( r ) /h ′ ( r ) . Then S ≤ lim inf r → f ( r ) − f (0) h ( r ) ≤ lim sup r → f ( r ) − f (0) h ( r ) ≤ S. In particular, if S = S , i.e. if the limit S := lim r → f ′ ( r ) /h ′ ( r ) ∈ [0 , ∞ ] exists then lim r → ( f ( r ) − f (0)) /h ( r ) exists as well and equals S . Proposition 3.1 yields the following general relations between generalized Minkow-ski contents and S-contents.
Theorem 3.2.
Let A ⊂ R d be bounded with V A (0) = 0 and let h : (0 , ∞ ) → (0 , ∞ ) be a differentiable function with lim r → h ( r ) = 0 . Assume that h ′ is nonzero onsome right neighbourhood of . Then, S ( h ′ ; A ) ≤ M ( h ; A ) ≤ M ( h ; A ) ≤ S ( h ′ ; A ) . Proof.
Apply Proposition 3.1 to f ( r ) := V A ( r ). (cid:3) JAN RATAJ AND STEFFEN WINTER
Now we establish bounds for generalized S-contents in terms of generalizedMinkowski contents providing the natural counterpart to the above inequalities.Taking into account the results of the previous section, it seems reasonable to as-sume that both the upper and lower generalized Minkowski content are positive andbounded - at least for the derivation of the lower bound. Again, the inequalitiesare derived from a more general statement on Kneser functions. Unfortunately, theresults below are restricted to gauge functions of the special form h ( r ) = r s g ( r )with s > g . Although this covers most gauge functionsappearing the literature, the interesting case s = 0 (appearing e.g. for the path ofBrownian motion in R d ) is excluded. Proposition 3.3.
Let f be a Kneser function of order d ≥ . If (3.1) 0 < lim inf r → f ( r ) h ( r ) ≤ lim sup r → f ( r ) h ( r ) < ∞ for some function h : (0 , ∞ ) → (0 , ∞ ) of the form h ( r ) = r s g ( r ) where s ∈ (0 , ∞ ) and g is non-decreasing, then < lim inf r → f ′ + ( r ) r − h ( r ) ≤ lim sup r → f ′− ( r ) r − h ( r ) < ∞ . Moreover, if additionally g (and thus h ) is differentiable and lim sup r → rg ′ ( r ) g ( r ) < ∞ ,then < lim inf r → f ′ + ( r ) h ′ ( r ) ≤ lim sup r → f ′− ( r ) h ′ ( r ) < ∞ . Proof.
By the assumption there exist 0 < m < M < ∞ and r > r ∈ (0 , r ), mh ( r ) ≤ f ( r ) ≤ M h ( r ) . (3.2)Choose a > ma s − M > ar ≤ r , f ( ar ) − f ( r ) ≥ m ( ar ) s g ( ar ) − M r s g ( r ) ≥ ( ma s − M ) r s g ( r ) , since g is non-decreasing. Combining this with inequality (2.1) (which still holds,since f is a Kneser function), we obtain f ′ + ( r ) r s − g ( r ) ≥ da d − ma s − M ) > r ∈ (0 , r a ), and thus lim inf r → f ′ + ( r ) r − h ( r ) >
0. If additionally g (and thus h ) isdifferentiable, then h ′ ( r ) = sr s − g ( r ) + r s g ′ ( r ) = r s − g ( r )( s + rg ′ ( r ) g ( r ) ). Hencelim inf r → f ′ + ( r ) h ′ ( r ) = lim inf r → f ′ + ( r ) r s − g ( r ) · s + rg ′ ( r ) g ( r ) ≥ lim inf r → f ′ + ( r ) r s − g ( r ) · lim inf r → s + rg ′ ( r ) g ( r ) , where the first lim inf is positive as we have just shown and for the second one thisfollows from the assumption lim sup r → rg ′ ( r ) g ( r ) < ∞ .Again a similar but slightly simpler argument shows the finiteness of the corre-sponding limes superior. Choosing some 0 < b <
1, we get from (3.2) f ( r ) − f ( br ) ≤ f ( r ) ≤ M h ( r ) = M r s g ( r ) . HARACTERIZATION OF MINKOWSKI MEASURABILITY 9
Combining this with inequality (2.4) (which holds, since f is a Kneser function),we obtain f ′− ( r ) r s − g ( r ) ≤ Md − b d , which implies the finiteness of the first lim sup. If again, g (and thus h ) is differentiable, we havelim sup r → f ′− ( r ) h ′ ( r ) = lim sup r → f ′− ( r ) r s − g ( r ) · s + rg ′ ( r ) g ( r ) ≤ lim sup r → f ′− ( r ) r s − g ( r ) · lim sup r → s + rg ′ ( r ) g ( r ) . The expression in the last lim sup is trivially bounded from above by 1 /s and thuswe conclude the finiteness of lim sup r → f ′− ( r ) h ′ ( r ) . Note that the assumption on thelimes inferior is not needed for the proof of the bounds for the lim sup’s and thatalso the last condition lim sup r → rg ′ ( r ) g ( r ) < ∞ is not used in this part. Since b can bechosen arbitrarily close to 0 and M arbitrarily close to lim sup r → f ( r ) h ( r ) , we obtainthe relation lim sup r → f ′− ( r ) r − h ( r ) ≤ d lim sup r → f ( r ) h ( r ) . (3.3) (cid:3) Theorem 3.4.
Let A be a bounded subset of R d such that < M ( h ; A ) ≤ M ( h ; A ) < ∞ , for some gauge function h : (0 , ∞ ) → (0 , ∞ ) , r r d − D g ( r ) with D ∈ [0 , d ) > and g non-decreasing. Then < S ( r − h ; A ) ≤ S ( r − h ; A ) < ∞ . If additionally, g is differentiable and lim sup r → rg ′ ( r ) g ( r ) < ∞ , then < S ( h ′ ; A ) ≤ S ( h ′ ; A ) < ∞ . Proof.
Let f ( r ) := V A ( r ) and recall that f is a Kneser function of order d for eachbounded set A in R d . Set s := d − D . The assumptions imply that s > < lim inf r → ( V A ) ′ + ( r ) r d − D − g ( r ) ≤ lim sup r → ( V A ) ′− ( r ) r d − D − g ( r ) < ∞ . Now note that S A ( r ) ≥ ( V A ) ′ + ( r ) for each r >
0, which implies the positiv-ity of S ( r − h ; A ), and that S A ( r ) ≤ ( V A ) ′− ( r ) for each r >
0, which allowsto conclude the finiteness of S ( r − h ; A ). If g (and thus h ) is differentiable andlim sup r → rg ′ ( r ) g ( r ) < ∞ , then the second assertion of Proposition 3.3 can be applied.Again, the inequality S A ( r ) ≥ ( V A ) ′ + ( r ), yields the positivity of S ( h ′ ; A ) and theinequality S A ( r ) ≤ ( V A ) ′− ( r ) the finiteness of S ( h ′ ; A ). (cid:3) Remark 3.5.
Note that for the finiteness of S ( r − h ; A ) (and S ( h ′ ; A ), respec-tively), only the finiteness of M ( h ; A ) is required, while for the corresponding lowerbounds the whole hypothesis is needed. Moreover, as a corollary to the proof ofProposition 3.3 we obtain the direct relation S ( r − h ; A ) ≤ d M ( h ; A ) , and, in case g is differentiable, also S ( h ′ ; A ) ≤ d M ( h ; A ). Refining the argument in the proof of Proposition 3.3, we will now establish,that the existence of the generalized Minkowski content M ( h ; A ) of a bounded set A ⊂ R d (of Minkowski dimension D ∈ [0 , d )) implies the existence of its generalizedS-content S ( h ′ ; A ). This extends Proposition 2.3 and its corollaries to the case ofgauge functions. Proposition 3.6.
Let f be a Kneser function of order d ≥ . If (3.4) lim r → f ( r ) h ( r ) = C for some function h : (0 , ∞ ) → (0 , ∞ ) of the form h ( r ) = r s g ( r ) where s, C ∈ (0 , ∞ ) are positive constants and g is non-decreasing, then lim r → f ′ + ( r ) sr s − g ( r ) = lim r → f ′− ( r ) sr s − g ( r ) = C. If g is differentiable and lim r → rg ′ ( r ) g ( r ) = 0 , then lim r → f ′ + ( r ) h ′ ( r ) = lim r → f ′− ( r ) h ′ ( r ) = C. Proof.
Let ε ∈ (0 , min { , C } ). Choose a > a s ≥ C + √ εC − √ ε . Since √ ε > ε and thus C + √ ε > C + ε and C − √ ε < C − ε , we have( C − ε ) a s − ( C + ε ) > . Repeating the argument of the first part of the proof of Proposition 3.3 with m := C − ε and M := C + ε , we infer that there exists some r = r ( ε ) such that f ′ + ( r ) r s − g ( r ) ≥ da d − C − ε ) a s − ( C + ε )) = dC a s − a d − − dε a s + 1 a d − > r ∈ (0 , r ). Hencelim inf r → f ′ + ( r ) r s − g ( r ) ≥ dC a s − a d − − dε a s + 1 a d − ε ∈ (0 , min { , C } ) and each a > a suchthat equality holds in (2.9) and let ε tend to 0. Then a = a ( ε ) converges to 1 andso does a t for each t ∈ R . Moreover, for the last term on the right hand side wehave lim sup ε → dε a s + 1 a d − ≤ lim ε → dε (cid:16) C + √ εC −√ ε (cid:17) ds − , which is easily seen by applying L’Hˆopital’s rule. Using again L’Hˆopital’s rule, weconclude thatlim inf r → f ′ + ( r ) sr s − g ( r ) ≥ ds C lim a ց a s − a d − ds C lim a ց sa s − da d − = C .
An analogous argument shows thatlim sup r → f ′− ( r ) sr s − g ( r ) ≤ C. HARACTERIZATION OF MINKOWSKI MEASURABILITY 11 (Here one can choose b = b ( ε ) such that b s = ( C − √ ε ) / ( C + √ ε ) and derive theinequality lim sup r → f ′− ( r ) r s − g ( r ) ≤ dC − b s − b d + dε b s − b d from which the claim follows by letting again ε → f ′ + ≤ f ′− . Now assume that g (and thus h ) is differentiable and that lim r → rg ′ ( r ) g ( r ) =0. Since h ′ ( r ) = r s − g ( r )( s + rg ′ ( r ) g ( r ) ), we get on the one handlim inf r → f ′ + ( r ) h ′ ( r ) = lim inf r → f ′ + ( r ) sr s − g ( r ) ·
11 + s rg ′ ( r ) g ( r ) ≥ C · lim inf r →
11 + s rg ′ ( r ) g ( r ) = C, and on the other handlim sup r → f ′− ( r ) h ′ ( r ) = lim sup r → f ′− ( r ) sr s − g ( r ) ·
11 + s rg ′ ( r ) g ( r ) ≤ C · lim sup r →
11 + s rg ′ ( r ) g ( r ) = C. (cid:3) Theorem 3.7.
Let A be a bounded subset of R d and let h : (0 , ∞ ) → (0 , ∞ ) be agauge function of the form h ( r ) = r d − D g ( r ) for some D ∈ [0 , d ) and some non-decreasing, differentiable function g satisfying lim r → rg ′ ( r ) g ( r ) = 0 . Let M ∈ (0 , ∞ ) .Then M ( h ; A ) = M, if and only if S ( h ′ ; A ) = M. That is, the set A is generalized Minkowski measurable (with respect to h ) if and onlyif it is generalized S-measurable (with respect to h ′ ) and in this case both contentscoincide.Proof. Assume that M ( h ; A ) = M . Let f ( r ) := V A ( r ) and s := d − D and applyProposition 3.6 (for which the assumption lim r → rg ′ ( r ) g ( r ) = 0 used). Then S ( h ′ ; A ) = M follows from the fact that ( V A ) ′ + ≤ S A ≤ ( V A ) ′− . The reverse implication is adirect consequence of Theorem 3.2. The special form of h and the fact that g is nondecreasing imply lim r → h ( r ) = 0 and g ′ ( r ) ≥
0. The latter implies alsothat h ′ ( r ) = sr s − g ( r ) + r s g ′ ( r ) is non-zero for r >
0. Hence the hypotheses ofTheorem 3.2 are satisfied. (cid:3)
Remark 3.8.
The condition lim r → rg ′ ( r ) g ( r ) = 0 in the statement above is a veryreasonable assumption. Roughly it means that the function g grows slower thanany power r t , t > r t ) ′ = tr t − and so r · ( r t ) ′ r t = t ). Thus a violation of thisassumption essentially means that the exponent s in h ( r ) = r s g ( r ) is not chosencorrectly. The condition lim sup r → rg ′ ( r ) g ( r ) < ∞ appearing in the statements beforeis even weaker. Note that if the function g is e.g. concave, then we have rg ′ ( r ) g ( r ) ≤ r > h is also not really a restriction, since we are onlyinterested in the asymptotics as r →
0. For any continuous, non-decreasing function h : (0 , ∞ ) → (0 , ∞ ) there is a differentiable function ˜ h : (0 , ∞ ) → (0 , ∞ ) such thatlim r → h ( r ) h ( r ) = 1. Counterexamples
In [1], an example of a Kneser function f was presented with the propertylim r → f ( r ) h ( r ) = 1 (shortly, f ( r ) ∼ h ( r ) as r → + )for the gauge function h ( x ) = | log x | − , while at the same time the limitlim r → + f ′ ( r ) /h ′ ( r ) did not exist. We recall this construction here and modify itin order to get a Kneser function f for which even lim inf r → + f ′ ( r ) /h ′ ( r ) = 0and lim sup r → + f ′ ( r ) /h ′ ( r ) = ∞ holds. The examples show in particular, that theassumptions on the gauge function h in Propositions 3.3 and 3.6 cannot be droppedin general.Consider the following scheme producing Kneser functions of order 2. Let r i ց a i ր ∞ be two monotone positive sequences such that(4.1) X i a i r i < ∞ . Let f be such that f ′ ( x ) = 2 a i x, x ∈ ( r i +1 , r i ) , i = 1 , , . . . . Condition (4.1) guarantees that such an f exists, namely, f ( x ) = X j ≥ i Z r j r j +1 a j y dy + Z xr i a i y dy = X j ≥ i a j ( r j − r j +1 ) + Z xr i a i y dy, x ∈ ( r i − , r i ) . The monotonicity of ( a i ) ensures that f is a Kneser function of order 2.In [1], the authors considered (up to some constant) the following case: r i = 2 − i , a i = 2 i i ( i + 1) , i = 1 , , . . . . It is not difficult to verify that f ( r i ) = X j ≥ i a j ( r j − r j +1 ) = 34 X j ≥ i j ( j + 1) = 34 i = 3 log 24 1 | log r i | and, choosing h ( r ) = | log r | , from the monotonicity of f we get f ( r ) ∼ h ( r ) , as r → . On the other hand, we have f ′ ( r i − ) = 2 a i r i = 2 2 i i ( i + 1) = 2 ii + 1 (log 2) r i (log r i ) = 8 log 23 ii + 1 h ′ ( r i ) ,f ′ ( r i +) = 2 a i − r i = 2 2 i i ( i + 1) = 12 ii + 1 (log 2) r i (log r i ) = 2 log 23 ii + 1 h ′ ( r i ) . Consequently, lim inf r → + f ′ ( r ) h ′ ( r ) ≤ < ≤ lim sup r → + f ′ ( r ) h ′ ( r ) . This shows that in Proposition 3.6 the assumption h ( r ) = r s g ( r ) with some s > g non-decreasing cannot be omitted . HARACTERIZATION OF MINKOWSKI MEASURABILITY 13
We shall now consider a modified version of the above example showing thatalso in Proposition 3.3 the assumption on the gauge function cannot be relaxed.Consider the sequences r i = 2 − i , a i = 2 i +1 i ( i + 1)which are again monotone. The associated function f is a Kneser function of order2. Again, condition (4.1) is clearly fulfilled. We have f ( r i ) = X j ≥ i a j ( r j − r j +1 ) = X j ≥ i j ( j + 1) (cid:16) − − j +1 (cid:17) ∼ X j ≥ i j ( j + 1) = 1 i as i → ∞ . Thus, with the gauge function h ( r ) = log 2 / log | log r | , since h ( r i ) = log 2 i log 2 + log log 2 ∼ i , we get f ( r i ) ∼ h ( r i ) as i → ∞ , and, using the monotonicity of f , it is not difficultto see that f ( r ) ∼ h ( r ) , r → + . On the other hand, we have h ′ ( r i ) = 1log i − i i , whereas f ′ ( r i − ) = 2 a i r i = 2 2 i i ( i + 1) ,f ′ ( r i +) = 2 a i − r i = 2 1 i ( i + 1) . Thus, 0 = lim inf r → + f ′ ( r ) h ′ ( r ) < lim sup r → + f ′ ( r ) h ′ ( r ) = ∞ . The counterexamples presented above are examples of Kneser functions. It is notclear whether there exist sets, having these functions as their volume function. Itwould be even more interesting, to obtain analogous examples of volume functionsof sets. This seems to be a more difficult problem and we formulate it here as anopen question.
Question 4.1.
Does there exist a bounded set A ⊂ R n and a gauge function h (assumed to be differentiable without loss of generality) such that A is generalizedMinkowski measurable with respect to h , but not generalized S-measurable with re-spect to h ′ ? Subsets of the line
We draw our attention to the case d = 1 and relate the above results to theresults of Lapidus and Pomerance in [8], where a characterization of Minkowskimeasurability is given in connection with the proof of the Modified Weyl-Berryconjecture in dimension one. In view of the previous sections, it is natural to addthe criterion of S-measurability to the equivalent characterizations of Minkowskimeasurability given. It turns out that, using the S-content as an intermediatestep and applying the above results, some of the proofs in [8] can be significantlysimplified. Moreover, the results below indicate that from a certain point of view,the surface area of the parallel sets might be the proper object in higher dimensionsto replace the fractal string associated to sets on the real line.Recall that to any compact subset F ⊂ R , one can associate a unique fractalstring L = ( l j ) ∞ j =1 , that is, a nonincreasing sequence of real numbers l j encoding thelengths of the bounded complementary intervals of F . When comparing the formu-las below with the ones in [8], it should be kept in mind that here we have an addi-tional normalization constant κ d − s in the definition of the s -dimensional Minkowskicontent. In analogy with the notation in the previous section, for two positive se-quences ( a j ) j ∈ N and ( b j ) j ∈ N we write a j ∼ b j as j → ∞ , if lim j →∞ a j /b j = 1,and similarly a j ≈ b j as j → ∞ , if there are constants c, C and j such that c ≤ a j /b j ≤ C for all j ≥ j . Theorem 5.1.
Let F ⊂ R be a compact set with Minkowski dimension dim M F = D ∈ (0 , (i.e. in particular with λ ( F ) = 0 ) and let L = ( l j ) ∞ j =1 be the fractal stringassociated with F . (a) Two sided bounds. The following assertions are equivalent: (i) 0 < M D ( F ) ≤ M D ( F ) < ∞ (ii) 0 < S D ( F ) ≤ S D ( F ) < ∞ (iii) l j ≈ j − /D as j → ∞ (b) Criterion of Minkowski measurability. The following assertions are equivalent: (i) F is Minkowski measurable, (ii) F is S-measurable, i.e., < S D ( F ) < ∞ , (iii) l j ∼ Lj − /D as j → ∞ for some L > .Under these latter assertions, Minkowski and S-content of F are given by M D ( F ) = S D ( F ) = 2 − D κ − D L D − D . (5.1)
Proof. (a) The equivalence ( i ) ⇔ ( ii ) is the case d = 1 of Theorem 2.2. Theequivalence ( i ) ⇔ ( iii ) is part of Theorem 2.4 in [8]. However, we give a simplerdirect proof of the equivalence ( ii ) ⇔ ( iii ), connecting the S-content directly to theasymptotics of the lengths l j in the associated fractal string L .For a proof of the implication ( iii ) ⇒ ( ii ), recall that ( iii ) means there existconstants c , c , j such that c < j · l Dj < c for all j ≥ j . For easier comparison,we follow the notation in [8] and let α := lim inf j →∞ l j j /D and β := lim sup j →∞ l j j /D , (5.2) HARACTERIZATION OF MINKOWSKI MEASURABILITY 15 cf. [8, (3.5), p.48]. Obviously, 0 < α ≤ β < ∞ . Furthermore, let J ( ε ) := max { j ∈ N : l j ≥ ε } , ε >
0. For 2 r ∈ ( l j +1 , l j ], we have J (2 r ) = j and thus H ( ∂F r ) = 2 + 2 J (2 r ) = 2 + 2 j. (5.3)Hence 2 − D l Dj +1 ( j + 1) ≤ r D H ( ∂F r ) ≤ − D l Dj ( j + 1) . We conclude that on the one hand κ − D S D ( F ) = lim inf r ց H ( ∂F r )(1 − D ) r − D ≥ − D − D lim inf j →∞ l Dj +1 ( j + 1) = 2 − D − D α D > κ − D S D ( F ) = lim sup r ց H ( ∂F r )(1 − D ) r − D ≤ − D − D lim inf j →∞ l Dj j · j + 1 j = 2 − D − D β D < ∞ , cf. also [8, cf. Proof of Theorem 3.1, equation (3.9)]).For a proof of the reverse implication ( ii ) ⇒ ( iii ), we essentially employ theargument in [8, Lemma 3.6], which gives the lower bound, and observe that a similarargument works for the upper bound: Letting α j := l j j /D , it is not difficult to seethat l j = l j +1 implies α j < α j +1 . This yields α = lim inf j : l j >l j +1 α j +1 and, similarly, β = lim sup j : l j >l j +1 α j . (5.4)Suppose ( ii ) holds. Then, in view of the relation H ( ∂F r ) = 2 + 2 J (2 r ), there existpositive constants r , m, M such that m ≤ r D J ( r ) ≤ M for all 0 < r ≤ r . For l j +1 < l j and r ∈ ( l j +1 , l j ] one has J ( r ) = j and thus m ≤ r D j ≤ M , which implies m /D j − /D ≤ r ≤ M /D j − /D . Letting r → l j +1 in this equation, we get on the one hand l j +1 ≥ m /D j − /D >m /D ( j + 1) − /D , which implies α j +1 > m /D for each j with l j > l j +1 . Thus α = lim inf j : l j >l j +1 α j +1 ≥ m /D > . On the other hand, we get for r = l j , l j ≤ M /D j − /D and so α j ≤ M /D for each j with l j > l j +1 . Hence β = lim sup j : l j >l j +1 α j ≤ M /D . (b) The equivalence ( i ) ⇔ ( ii ) is the case d = 1 of Theorem 2.4. The equivalence( i ) ⇔ ( iii ) is proved in [8, Theorems 3.1(b) and 4.1]. In particular, the latterTheorem has a long and technical proof. We prove the equivalence ( ii ) ⇔ ( iii )instead: The implication ( iii ) ⇒ ( ii ) follows immediately from the proof of thesame implication in (a) by setting α = β = L .For a proof of the reverse implication, we refine the argument of the correspond-ing proof in part (a). Assume S D ( F ) = 2 − D κ − D L D − D (5.5)for some L >
0, which implies that for each ε > r > L D − ε ≤ D − r D H ( ∂F r ) ≤ L D + ε for 0 < r ≤ r . It suffices to show α ≥ L and β ≤ L , where α and β are as in (5.2).Recalling that H ( ∂F r ) = 2 + 2 J (2 r ) and substituting t = 2 r , we infer L D − ε ≤ t D (1 + J ( t )) ≤ L D + ε for all 0 < t ≤ r . Now, if l j +1 < l j and t ∈ ( l j +1 , j j ], then J ( t ) = j and so( L D − ε ) /D ≤ t (1 + j ) /D ≤ ( L D + ε ) /D . (5.6)Setting t = l j and taking into account (5.4), we concludelim sup j →∞ l j j /D ≤ lim sup j : l j >l j +1 l j ( j + 1) /D ≤ ( L D + ε ) /D for each ε > ε → β ≤ L . Similarly, by letting t → l j +1 in(5.6) and using again 5.4, we getlim inf j →∞ l j j /D = lim inf j : l j >l j +1 l j +1 ( j + 1) /D ≥ ( L D − ε ) /D and so, by letting ε → α > L . This completes the proof of the implication( ii ) ⇒ ( iii ) in part (b) and thus of Theorem 5.1. (cid:3) Remark 5.2. ( Connection to the Modified Weyl-Berry (MWB) conjecture ) LetΩ = F c ∩ conv( F ) be the bounded open set consisting of the bounded complemen-tary intervals of the compact set F . In [8], the Minkowski content is connected tothe following function δ : (0 , ∞ ) → (0 , ∞ ), which describes the error if one tries topack intervals of a fixed small length l = 1 /x into the complementary intervals of F (whose lengths are given by the associated fractal string L = ( l j ) ∞ j =1 ): δ ( x ) = ∞ X j =1 l j x − ∞ X j =1 [ l j x ] = ∞ X j =1 { l j x } . (5.7)Here [ z ] and { z } denote the integer part and the fractional part of a number z ∈ R ,respectively.According to [8, Theorem 2.4], the following assertion is equivalent to the items( i ) − ( iii ) in part (a) of Theorem 5.1: δ ( x ) ≈ x D as x → ∞ . Similarly, by [8, Theorem 4.2], the assertions in part (b) of Theorem 5.1 imply that δ ( x ) ∼ − ζ ( D ) L D x D , as x → ∞ , (5.8)where ζ denotes the Riemann zeta-function. This relation is the key to the proofof the MWB conjecture in dimension one in [8], which connects the geometry ofthe set Ω to its spectral properties (that is, to its sound). Let λ ≤ λ ≤ . . . bethe eigenvalues of the Dirichlet Laplacian ∆ on Ω in increasing order and countedaccording to their multiplicities and let N ( λ ) := { k ≥ λ i ≤ λ } be the eigen-value counting function of ∆. The MWB conjecture states that the second orderasymptotic behavior of N ( λ ) is governed by the Minkowski content of the boundary F = ∂ Ω of Ω. (The first order asymptotics is well known to be given by the socalled
Weyl term ϕ ( λ ) involving the volume of Ω.) More precisely, for d = 1, onehas N ( λ ) = ϕ ( λ ) − c ,D M D ( F ) λ D/ + o ( λ D/ ) as λ → ∞ , HARACTERIZATION OF MINKOWSKI MEASURABILITY 17 where ϕ ( λ ) = π − V (Ω) λ / and c ,D = 2 D − π − D κ − D (1 − D ) ζ ( D ). This is easilyseen from (5.8) and the relation ϕ ( λ ) − N ( λ ) = ∞ X j =1 l j x − ∞ X j =1 [ l j x ] = δ ( x )where x = √ λ/π . We refer to [8] for more details on the resolution of the MWBconjecture in dimension one and to [9] for its disproof in higher dimensions, seealso [6]. Surprisingly, a certain converse of the implication above connecting (5.8)to the assertions in Theorem 5.1(b) is not true in the case D = and it is true forany other value D ∈ (0 ,
1) if and only if the Riemann hypothesis is true, as derivedby Lapidus and Maier in [7].
Remark 5.3. ( One sided bounds ) Under the hypothesis of Theorem 5.1, also thefollowing equivalence for the upper bounds holds regardless of any conditions onthe lower bounds:( i ) M D ( F ) < ∞ ⇔ ( ii ) S D ( F ) < ∞ ⇔ ( iii ) β < ∞ . Indeed, this is obvious from the proofs of Theorem 5.1(a) and Proposition 2.1, forthe equivalence of (i) and (iii) see also [8, Theorem 3.10]. In the latter paper, it wasalso observed that the corresponding equivalence is not true for the lower bounds: α > M D ( F ) > S D ( F ) > M D ( F ) > S D ( F ) > ⇔ α > Remark 5.4. ( Generalized contents of subsets of R ) For the generalized contentsa statement completely analogous to Theorem 5.1 can be derived by combining theresults in Section 3 with the results of He and Lapidus in [2, 3], see in particularTheorems 2.4 and 2.6 in [2] (or Theorems 2.5 and 2.7 in [3]), where correspondingequivalent assertions for the S-content can be added. References [1] O. Honzl, J. Rataj: Almost sure asymptotic behaviour of the r-neighbourhood surface areaof Brownian paths. To appear in
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