Oscillatory Integrals and Fractal Dimension
OOSCILLATORY INTEGRALS AND FRACTAL DIMENSION
J.-P. ROLIN, D. VLAH, V. ˇZUPANOVI ´C
Abstract.
We study geometrical representation of oscillatory integralswith an analytic phase function and a smooth amplitude with compactsupport. Geometrical properties of the curves defined by the oscillatoryintegral depend on the type of a critical point of the phase. We giveexplicit formulas for the box dimension and the Minkowski content ofthese curves. Methods include Newton diagrams and the resolution ofsingularities.
Keywords: oscillatory integral, box dimension, Minkowski content, criti-cal points, Newton diagram.AMS Classification: 58K05, 42B20, (secondary 28A75, 34C15)1.
Introduction, motivation and definitions
This paper is a starting point of a study intended to relate the stan-dard classification of singularities of maps with the fractal dimension andthe Minkowski content of curves defined by oscillatory integrals. The closelink between the theory of singularities and the investigation of oscillatoryintegrals is well-known, and is explained in detail in [2]. Our purpose is toconnect these notions to the analysis of fractal data of curves as it is de-scribed in [14]. In particular we consider the box counting dimension (alsocalled the box dimension ), and the
Minkowski content . It is worth noticingthat every rectifiable curve has a box dimension equal to 1. Hence the boxdimension is a tool to distinguish nonrectifiable curves. Notice that anothercommonly used fractal dimension, the Hausdorff dimension, which takes thevalue 1 on every non rectifiable smooth curve, cannot distinguish betweenthem.One motivation originates in previous works, in which the behavior of a(discrete or continuous) dynamical system in the neighborhood of a singularpoint is analyzed through the box dimension of an orbit. For example, in[16], the authors consider a family of planar polynomial vector fields, calledthe standard model of the Hopf-Takens bifurcation . They prove that the boxdimension of any trajectory spiraling in the neighborhood of a limit cycle ofmultiplicity m has the box dimension 2 − /m . They also link in [17], fora planar analytic system with a weak focus singular point, the box dimen-sion of a spiraling trajectory and the Lyapunov coefficients of the singularity. a r X i v : . [ m a t h . C A ] S e p J.-P. ROLIN, D. VLAH, V. ˇZUPANOVI´C
If we consider now a discrete dynamical system on the real line in theneighborhood of a fixed point, the box dimension of a discrete orbit is re-lated to the multiplicity of the generating function. This approach, togetherwith the standard methods combining the study of discrete and continuoussystems via the use of Poincar´e first return map, leads to further results (see[10] and [17]).It is proved in [12] that the formal class of an analytic parabolic diffeo-morphism is fully determined by the knowledge of a fractal data of a singleorbit: namely, its box dimension, its Minkowski content and another num-ber called its residual content .Based on these considerations, it seems relevant to study the singularitiesof a map f : R n → R n by considering the fractal data of an oscillatoryintegral with a phase f , and its geometric representation as a plane curveparametrized by its real and imaginary part. We actually observed a rela-tion between the type of a critical point of the phase and the box dimensionof the associated curve: a “high degeneracy” of the critical point causes a“big accumulation” of the curve, which is reflected by a larger box dimen-sion. This is the exact analogue of the phenomenon observed above for theorbits or trajectories of dynamical systems. A well-known example of thissituation is the oscillatory Fresnel integral , and its geometric representation,the
Cornu spiral (also known as clothoid or Euler spiral ). This curve playsan important role in the problem of the construction of optimal trajectoriesof a planar motion with a bounded derivative of the curvature; see [8]. Itsfractal data have been computed in [6]. It is worth noticing that the phasefunction of a Fresnel integral has only non-degenerate critical points.Our results can be summarized as follows. We consider oscillatory in-tegrals with an analytic phase function and an amplitude with compactsupport. We study the graph of the oscillatory integrals I ( τ ), as τ → ∞ ,and also the curves defined in a standard way, analogously as the Cornuspiral, which are defined by the parametrization given by the real and imag-inary parts of the integral I ( τ ). We show that the box dimension and theMinkowski content of the curves reveal the leading term of the asymptoticexpansion. More precisely, the oscillation index can be read from the boxdimension, while the leading coefficient can be read from the Minkowskicontent in the case of Minkowski nondegeneracy. Minkowski degeneracycorresponds to a nontrivial multiplicity of the oscillation index. In particu-lar, for phase functions of two variables, we show explicitly how to connectthese notions to their Newton diagram.We plan to pursue the present work in various directions. One goal isthe study, from our point of view, the bifurcations in parametric familiesof maps and their caustics. Second, we would like to know how our resultsbehave if we take, in the oscillatory integral, an amplitude function which isnot of class C ∞ (for example, oscillatory integrals on halfspaces). Finally, SCILLATORY INTEGRALS AND FRACTAL DIMENSION 3 we want to develop our subject in the direction of tame, but non-analyticphase functions.The main results of this paper are presented in three theorems, withrespect to the dimension of the space: Theorems 1, 2 and 3, for n = 1, n = 2 and n >
2, respectively. The main difference between the first twotheorems is caused by logarithmic terms which can appear in the expansionof the integral in Theorem 2, while in Theorem 1, that is not possible. InTheorem 3 powers of logarithmic terms can also appear.1.1.
The box dimension.
For A ⊂ R N bounded we define the ε -neighbour-hood of A as: A ε := { y ∈ R N : d ( y, A ) < ε } . By the lower s -dimensionalMinkowski content of A , for s ≥
0, we mean M s ∗ ( A ) := lim inf ε → | A ε | ε N − s , and analogously for the upper s -dimensional Minkowski content M ∗ s ( A ). If M ∗ s ( A ) = M s ∗ ( A ), we call the common value the s -dimensional Minkowskicontent of A , and denote it by M s ( A ). The lower and upper box dimensionsof A are dim B A := inf { s ≥ M s ∗ ( A ) = 0 } and analogously dim B A := inf { s ≥ M ∗ s ( A ) = 0 } . If these two valuescoincide, we call it simply the box dimension of A , and denote it by dim B A .This will be our situation. If 0 < M d ∗ ( A ) ≤ M ∗ d ( A ) < ∞ for some d ,then we say that A is Minkowski nondegenerate . In this case obviously d = dim B A . In the case when the lower or upper d -dimensional Minkowskicontent of A is equal to 0 or ∞ , where d = dim B A , we say that A is degenerate . If there exists M d ( A ) for some d and M d ( A ) ∈ (0 , ∞ ), then wesay that A is Minkowski measurable . For more details on these definitionssee, e.g., Falconer [4], and [16].1.2.
Examples of the box dimension. (1) A basic example of fractal sets with a nontrivial box dimension isthe a -string defined by A = { k − a : k ∈ N } , where a >
0, introducedby Lapidus; see, e.g., [9]. Here is dim B A = 1 / (1 + a ).(2) Furthermore, important examples are curves from Tricot’s formulas;see [14, p. 121]. The box dimension of a spiral in the plane defined inthe polar coordinates by r = m ϕ − α , ϕ ≥ ϕ >
0, where ϕ , m > α ∈ (0 ,
1] are fixed, is equal to 2 / (1 + α ).(3) Assuming that 0 < α ≤ β , the box dimension of the graph of thefunction f α,β ( x ) = x α sin( x − β ), for x ∈ (0 , α, β )-chirp, is equal to 2 − ( α + 1) / ( β + 1); see [14, p. 121]. J.-P. ROLIN, D. VLAH, V. ˇZUPANOVI´C
Oscillatory integrals.
One of the main objects of interest in thispaper are the oscillatory integrals(1) I ( τ ) = (cid:90) R n e iτf ( x ) φ ( x ) dx, τ ∈ R , where f is called the phase function and φ the amplitude.Throughout this paper in all theorems we will use the following assump-tions on the phase function f and the amplitude φ that we call the standardassumptions . The amplitude function φ : R n → R , • is of class C ∞ , • is a non-negative function with compact support, • the point 0 ∈ R n is contained in the interior of the support of thefunction φ .The phase function f : R n → R : • the point 0 is a critical point of the function f , • f is a real analytic function in the neighborhood of its critical point 0, • the point 0 is the only critical point of the function f in the interiorof the support of the function φ .The asymptotic expansion of I ( τ ), as τ → ∞ , depends essentially on crit-ical points of f . The critical point of f is a point with all partial derivativesequal to zero. The nondegenerate critical point is a point were the Hessianis regular. In that case integral (1) is called the Fresnel integral in the ref-erence Arnold et all [2]. We use theorems from [2] to obtain the asymptoticexpansion of I ( τ ) as τ → ∞ , in the cases if f has no critical points, has thenondegenerate or the degenerate critical point. The phase function f deter-mines exponents in the asymptotic expansion, while the amplitude functiondetermines the coefficients. We will discuss curves defined by the oscillatoryfunctions X ( τ ) = Re I ( τ ) ,Y ( τ ) = Im I ( τ ) , (2)for τ near ∞ , and also the reflected functions x ( t ) := X (1 /t ), y ( t ) := Y (1 /t ),as t → Oscillation and singular indices.
Applying [2, Theorem 6.3] on (1)we get the asymptotic expansion(3) I ( τ ) ∼ e iτf (0) (cid:88) α n − (cid:88) k =0 a k,α ( φ ) τ α (log τ ) k , as τ → ∞ . According to the same theorem, the parameter α is from the set consistingof a finite set of arithmetic progressions, which depend only on the phase φ ,and consisting of negative rational numbers. Coefficients a k,α depend onlyon the amplitude φ .The index set of an analytic phase f at a critical point is defined as the setof all numbers α having the property: for any neighborhood of the critical SCILLATORY INTEGRALS AND FRACTAL DIMENSION 5 point there is an amplitude with support in this neighborhood for which inthe asymptotic series (3) there is a number k such that the coefficient a k,α is not equal to zero. The oscillation index β of an analytic phase f at acritical point is the maximal number in the index set. The multiplicity of theoscillation index K of an analytic phase f at a critical point is the maximalnumber k having the property: for any neighborhood of the critical pointthere is an amplitude with support in this neighborhood for which in theasymptotic series (3) the coefficient a k,β is not equal to zero.The singular index of an analytic phase f in n variables at a critical pointis equal to β + n/
2. The multiplicity of the singular index is the multiplicityof β .1.5. Oscillatory and curve dimensions.
We say that x ( t ) = X (1 /t ) isoscillatory near the origin if X ( τ ) is oscillatory near τ = ∞ . We measurethe rate of oscillatority of X ( τ ) near τ = ∞ by the rate of oscillatority of x ( t ) near t = 0. More precisely, the oscillatory dimension dim osc ( X ) (near τ = ∞ ) is defined as the box dimension of the graph of x ( t ) near t = 0. Also,we investigate the associated Minkowski contents. Analogously for y ( t ) and Y ( τ ).Given the oscillatory integral I ( τ ) from (1), we define the curve dimension of I ( τ ) as the box dimension of the curve defined in the complex plane by I ( τ ), near τ = ∞ . As in the oscillatory dimension, we also investigate theassociated Minkowski contents. f ( x ) = x + 1 f ( x ) = x + 1 d = d = Figure 1.
Curves defined by oscillatory integrals I i ( τ ) from(1), for phase functions f i and their respective curve dimen-sions d i , see Theorem 1 below.It is well known that degenerate critical points of phase functions con-tribute to the leading term of the asymptotic expansion (3) of oscillatory J.-P. ROLIN, D. VLAH, V. ˇZUPANOVI´C integral (1). On the other hand, curve dimension of (2) will be determined bythe asymptotic expansion, so we will connect type of critical point with thecurve dimension. More precisely, in Theorems 2 and 3, the oscillatory andcurve dimensions are related to the oscillation index. Also, it is well knownthat the asymptotic expansion has been related to the Newton diagram ofthe phase function.1.6.
The Newton diagram.
According to [2], we will use the notion ofthe Newton polyhedron of the phase function to formulate our results inthe dimension n ≥
2. The Newton polyhedron is defined for the Taylorseries of the critical point. Let us consider the positive orthant of the space R n . We define the Newton polyhedron of an arbitrary subset of this orthantconsisting of points with integer coordinates. At all such points we take aparallel positive orthant. The Newton polyhedron is the convex hull in R n ofthe union of all parallel orthants mentioned above. The Newton diagram ∆of a subset is the union of compact faces of the Newton polyhedron of thesame subset.We consider the power series of the phase ff ( x ) = (cid:88) a k x k with real coefficients, having monomials x k = x k . . . x k n n with multi-index k = ( k , . . . k n ). The Newton polyhedron and diagram ofthis power series has been constructed using the multi-indices which are inthe reduced support of the series. Reduced support is obtained by remov-ing the origin from the support of the series. This support is a subset ofthe positive orthant, consisting of points having non-negative coordinates.These points are given by multi-indices of all monomials from the powerseries, having non-zero coefficients. The polynomial f ∆ that equals to thesum of monomials belonging to the Newton diagram, is called the princi-pal part of the series. To each face γ of the Newton diagram is associatedthe quasi-homogeneous polynomial. The type of quasi-homogeneity is de-termined by the slope of the face. Furthermore, we introduce the conceptof nondegeneracy of the principal part. Notice that in this article we have3 distinct types of nondegeneracy: • nondegeneracy of a critical point with respect to the Hessian, • nondegeneracy of the Minkowski content, • nondegeneracy of the principal part of the series.The principal part f ∆ of the power series f with real coefficients is R -nondegenerate if for every compact face γ of the Newton polyhedron of theseries the polynomials ∂f γ /∂x , . . . , ∂f γ /∂x n do not have common zeroes in ( R \ n . SCILLATORY INTEGRALS AND FRACTAL DIMENSION 7
Roughly speaking, R -nondegeneracy means that these mentioned deriva-tives have the same common zeroes as monomials. This property is essentialfor the resolution of the singularity, see [2, p. 195]. Furthermore, the set ofall series with a degenerate principal part is ‘small‘, more precisely, the setof R -nondegenerate series is dense in the space of all series with a fixedNewton polyhedron, see Lemma 6.1. [2]. A generalization of the notion ofthe principal part for R -degenerate vector fields could be found in [18].The asymptotic expansion of the oscillatory integrals is related to someproperties of critical points of its phase function, which could be read fromthe Newton diagram. Let us consider the bisector of the positive orthantin R n , that is the line consisting of points with equal coordinates. Thebisector intersects the boundary of the Newton polyhedron in the exactlyone point ( c, . . . , c ), which is called the center of the boundary of the Newtonpolyhedron. The number c is called the distance to the Newton polyhedron . Remoteness of the Newton polyhedron is equal to r = − /c . If r > − remote , which means that it does not contain thepoint (1 , . . . , Remoteness of the critical point of the phase is the upper boundof remotenesses of the Newton polyhedra of the Taylor series of the phasein all systems of local analytic coordinates with the origin at the criticalpoint. The coordinates in which the remoteness is the greatest, are calledthe adapted coordinates to the critical point.We consider the open face which contains the center of the boundary ofthe Newton polyhedron. The codimension of this face, less one, is called the multiplicity of the remoteness. If the face is a vertex then the multiplicity is n −
1, and if the face is an edge then the multiplicity is n − Main results
We use Theorems 6.1., 6.2., 6.3., 6.4. from [2] in order to measure theoscillatority of the oscillatory integral by using the box dimension. Thesetheorems give the asymptotic expansion of (1) if the phase f has no criticalpoints, nondegenerate and degenerate critical points. Theorem 6.4. involvesNewton diagrams. In our theorems we use these results about asymptoticexpansions.In Theorems 1, 2 and 3 we present our main results about fractal analysisof singularities in dimensions n = 1, n = 2 and n >
2, respectively. Proofsof these theorems are presented in Section 4.
Theorem 1 (The phase function of a single variable) . Let n = 1 , the stan-dard assumptions on f and φ hold, and let f (0) (cid:54) = 0 . Assume f (cid:48) (0) = f (cid:48)(cid:48) (0) = · · · = f ( s − (0) = 0 and f ( s ) (0) (cid:54) = 0 for some integer s ≥ . Let Γ be the curve defined by (1) and (2), near the origin. Then: ( i ) The oscillatory dimension of both X and Y from (2) is equal to d (cid:48) = s − s and associated graphs are Minkowski nondegenerate. J.-P. ROLIN, D. VLAH, V. ˇZUPANOVI´C ( ii ) The curve dimension of I is d = ss +1 , curve Γ is Minkowski measur-able, and d -dimensional Minkowski content of Γ is (4) M d (Γ) = | C | ss +1 · π · (cid:18) πs · f (0) (cid:19) − s +1 · s + 1 s − , where the constant C depends on the phase function f and on the amplitudefunction value in the origin φ (0) .Remark . The constant C can be explicitly calculated using a standardformula for phase functions with nondegenerate critical point; see Remark3 in Section 3 with examples. For a more general case of phase functions f see [13]. Theorem . Let n = 2 , the standardassumptions on f and φ hold, and let f (0) (cid:54) = 0 . Let β be the remoteness ofthe critical point of the phase function f . Let Γ be the curve defined by (1)and (2), near the origin, with asymptotic expansion (3). Then: ( i ) If the multiplicity of the remoteness β is equal to or the remoteness β is equal to − , then the oscillatory dimension of both X and Y from(2) is equal to d (cid:48) = ( β + 3) / and the associated graphs are Minkowskinondegenerate. The curve dimension of I is d = 2 / (1 − β ) and the associatedMinkowski content is (5) M d (Γ) = (cid:20) | a ,β ( φ ) | f (0) β (cid:21) − β · [ − β ] β − β · π β − β · − β β . ( ii ) If the multiplicity of the remoteness β is equal to and the remoteness β is bigger than − , then the oscillatory and curve dimensions are the sameas in the previous case with associated degenerate Minkowski contentsTheorem . Let n > thestandard assumptions on f and φ hold, and let f (0) (cid:54) = 0 . Let the principalpart of the Taylor series of f at its critical point is R -nondegenerate, andthe Newton polyhedron of this series is remote with the remoteness of theNewton polyhedron equal to β . Let Γ be the curve defined by (1) and (2),near the origin, having asymptotic expansion (3). Then: ( i ) If a ,β ( φ ) (cid:54) = 0 and a i,β = 0 , for i = 1 , . . . , n − , the oscillatorydimension of both X and Y from (2) is equal to d (cid:48) = ( β + 3) / and theassociated graphs are Minkowski nondegenerate. The curve dimension of I is d = 2 / (1 − β ) and the associated Minkowski content is given by (5). ( ii ) If for some
L > holds a L,β (cid:54) = 0 , the oscillatory and curve dimensionsare the same as for the previous case and the associated Minkowski contentsare degenerate.Remark . In Theorems 1, 2 and 3, if we take f (0) = 0, then the curve Γand the associated reflected graphs are rectifiable, and all dimensions areequal to 1. SCILLATORY INTEGRALS AND FRACTAL DIMENSION 9
If there are no singularities in the observed domain, which is given by thesupport of the amplitude φ , then Proposition 1 gives only a trivial fractaldimension. Proposition . (The regular phase function) Assume that the standard as-sumptions on φ hold, and that f does not have any critical point containedin the interior of the support of φ . Let Γ be the curve defined by (1) and(2), near the origin. Then Γ is a rectifiable curve and the curve dimensionof I is equal to . Furthermore, the graphs of the functions x ( t ) = X (1 /t ) and y ( t ) = Y (1 /t ) , where X and Y are from (2), are rectifiable. Hence, theoscillatory dimension of X and Y equals .Proof. From [2, Theorem 6.1] it follows that I ( τ ) tends to zero more rapidlythan any power of the parameter, as τ → + ∞ . The claim is based on theRiemann-Lebesgue lemma, see [15, p. 16]. For a 1-dimensional situation wehave I (cid:48) ( τ ) = i (cid:90) R e iτf ( x ) f ( x ) φ ( x ) dx. The integral I (cid:48) ( τ ) admits the same type of asymptotic expansion as I ( τ ) andall derivatives go to zero more rapidly than any power, that is, τ n I ( k ) ( τ ) → τ → + ∞ , for all k ≥ n ∈ N .We deduce that τ n (cid:113) X (cid:48) ( τ ) + Y (cid:48) ( τ ) → τ → + ∞ , for all n ∈ N , sothat Γ is rectifiable. Therefore (see [14]) its box dimension equals 1. For thesame reason x (cid:48) ( t ) = − t − X (cid:48) (1 /t ) → t →
0, hence (cid:82) x (cid:113) x (cid:48) ( t ) dt < ∞ . The same holds for y . It proves that graphs of functions x and y arerectifiable, so the oscillatory dimension of X and Y equals 1. (cid:3) Proposition 2 demonstrates that in dimensions higher than 2, nondegen-erate singularities cannot be detected by the fractal dimension.
Proposition . (The nondegenerate critical point in a higher dimension) Assume that the standard assumptions on φ and f hold, and that ∈ R n ,where n > , is a nondegenerate critical point of f (the Hessian matrix of f is not equal to zero). Let Γ be the curve defined by (1) and (2), nearthe origin. Then Γ is a rectifiable curve and the curve dimension of I isequal to . Furthermore, the graphs of the functions x ( t ) = X (1 /t ) and y ( t ) = Y (1 /t ) , where X and Y are from (2), are rectifiable. Hence, theoscillatory dimension of X and Y equals .Proof. The nondegeneracy of the critical point implies that I ( τ ) ∼ C · e iτf (0) · τ − n as τ → + ∞ , where C ∈ C (see [2, Theorem 6.2]. As in the proof ofProposition 1, I (cid:48) ( τ ) admits the same type of asymptotic expansion. Hence (cid:113) X (cid:48) ( τ ) + Y (cid:48) ( τ ) ≤ C τ − n for some C >
0, so Γ is rectifiable.As above, x (cid:48) ( t ) = t − X (cid:48) (cid:0) t (cid:1) ≤ C t n − for some C >
0. So (cid:113) x (cid:48) ( t ) ≤ C t n − for some C >
0. As n >
1, we conclude that the graph of x is rectifiable. The same holds for y . About the dimensions, we conclude as inthe proof of Proposition 1. (cid:3) Examples
Remark . In [2, Theorem 6.2] there is an explicit formula for the leadingcoefficient in the asymptotic expansion of the oscillatory integral with anondegenerate critical point of the phase in space of the dimension n . Ifthe phase f and the amplitude φ satisfy the standard assumptions, then aleading coefficient is the coefficient of the power τ − n/ and is equal to φ (0)(2 π ) n/ exp (cid:0) ( iπ/ · sgn ( f (cid:48)(cid:48) xx (0)) (cid:1) | det f (cid:48)(cid:48) xx (0) | − / . Example . A computation of the Minkowski content of the curve for thenondegenerate case in 1-dimensional space.Using Theorem 1, for s = 2 we obtain oscillatory and curve dimensionsfor the integral and the curve defined by (1) and (2), respectively. Theoscillatory dimension is equal to 5 / /
3. Using Remark 3, for n = 1 we compute C = φ (0) √ π | f (cid:48)(cid:48) (0) | − / exp (cid:0) ( iπ/ · sgn ( f (cid:48)(cid:48) (0)) (cid:1) , and using formula (4) we obtain the Minkowski content of the curve Γ M / (Γ) = 3 | C | π (cid:18) π f (0) (cid:19) − . For an example, if f ( x ) = x + 1, then we have M / (Γ) = 3 · / πφ (0) / . Example . A caustic consisting of the elementary critical points A k and D k .[1] and [2] introduced the classification of singularities using normal formsof singularities and parametric families. According to the assumptions of ourtheorems here we work with maps whose critical point does not coincide withthe zero point, so we shift the graph of our map. The situation when thesepoints coincide is not oscillatory, see expansion (3) for f (0) = 0, so we take f (0) = 1. Let us suppose that for a given value of the parameters, the phasefunction has a unique critical point. In this case the caustic in a neighbor-hood of the given value of the parameter is said to be elementary. Here wemention examples of elementary caustics obtained by varying two or threeparameters, [2, p. 174, 185], [1, p. 246]. The caustics consist of degeneratecritical points of type A k for k ≥
1, and D k for k ≥
4. Contributions of thecritical points of the phase to the asymptotic expansion of oscillatory inte-grals depend on the type of these critical points. Each degenerate criticalpoint has contribution of order τ γ − n/ , where γ = ( k − / (2 k + 2) for A k ,and γ = ( k − / (2 k −
2) for D k , which are singular indices. According toTheorem 2, the box dimension of the associated curve, the curve dimension,is equal to d = 2 / (1 − β ), where β = γ − n/
2. If k → ∞ then γ → /
2, so
SCILLATORY INTEGRALS AND FRACTAL DIMENSION 11 β → (1 − n ) /
2, hence the curve dimension d → / (1 + n ). We see that thecurve dimension increases and tends to 2 for n = 1, and tends to 4 / n = 2, when we have more complicated critical points whose singular indextends to 1 /
2. The oscillatory dimension is equal to d (cid:48) = β . Example . The normal forms of the type x p + y q .Consider the phase f ( x, y ) = x p + y q + 1, for integers p, q ≥ p, q ) (cid:54) =(2 , f (0 , (cid:54) = 0. In this case the remoteness β = − p − q , hence itfollows from Theorem 2 that the oscillatory dimension is equal to d (cid:48) = 21 + p + q , while the curve dimension is equal to d = 32 − p − q . The computation of Minkowski content (5) is more involved, as it dependon the computation of the first coefficient in asymptotic expansion (3) of theintegral. Notice that in this example we replace our standard notation Γ forthe curve associated to the oscillatory integral with C , in order to avoid aconfusion with the gamma function. According to [5] we can compute thefirst coefficient in the expansion. The phase is written in adapted coordi-nates, which means that the remoteness is the biggest possible, in the setof all remotenesses of Newton diagrams of the map in different coordinatesystems. In this case the Newton diagram has only one compact side S . Asthe bisector intersects the interior of the compact edge, the leading term ofthe asymptotic expansion is d ( φ ) τ − β , where β is the remoteness and φ theamplitude. First, define the function S ( x, y ) to be equal to f ( x, y ). Now,define the function S +0 ( x, y ) − d to be equal to S ( x, y ) − d when S ( x, y ) > S − ( x, y ) − d to be equalto ( − S ( x, y )) − d when S ( x, y ) < superadapted ; see [5], which means that S (1 , y ) and S ( − , y ) have noreal roots of order bigger than − /β , except y = 0. Hence, according to [5,Theorem 1.2], if we put c ( φ ) := φ (0 , m + 1 (cid:90) + ∞−∞ (cid:16) S +0 (1 , y ) β + S +0 ( − , y ) β (cid:17) dy,C ( φ ) := φ (0 , m + 1 (cid:90) + ∞−∞ (cid:16) S − (1 , y ) β + S − ( − , y ) β (cid:17) dy, where − /m is a slope of the edge S , then the leading term coefficient ofthe asymptotic expansion of I ( τ ) is equal to(6) a ,β ( φ ) = − β Γ ( − β ) (cid:16) e − i π β c ( φ ) + e i π β C ( φ ) (cid:17) . Obviously, there are four distinct cases in the computation regarding theintegers p and q being odd or even. We will compute the leading term coefficient and the Minkowski content (5) for the case of p and q beingeven. The other three cases are computed in similar fashion. We compute β = − /p − /q and m = p/q . After integration we obtain the result c ( φ ) = 4 φ (0 , q ( m + 1) B (cid:18) p , q (cid:19) , C ( φ ) = 0 ,a ,β ( φ ) = 4 φ (0 , e i π (cid:16) p + q (cid:17) Γ (cid:18) p + 1 (cid:19) Γ (cid:18) q + 1 (cid:19) , expressed using the beta function B and the gamma function Γ. As we took f (0 ,
0) = 1 and as we can without loss of generality fix φ (0 ,
0) = 1, we obtainthe Minkowski content of the curve C to be equal to M β ( C ) = (cid:20) (cid:18) p + 1 (cid:19) Γ (cid:18) q + 1 (cid:19)(cid:21) − β · [ − β ] β − β · π β − β · − β β , by putting the coefficient a ,β ( φ ) in formula (5), where β depends only on p and q . Notice that the Minkowski content depends essentially only on p and q .Finally, notice that the normal forms from this example include singulari-ties of the standard classification (see [1]) types E and E , for ( p, q ) = (3 , p, q ) = (3 , p, q ) = (2 , Proofs of main results
Proof of Theorem 1.
Without the loss of generality we assume that f (0) >
0. In the case of f (0) <
0, we consider the integral J having the phase (cid:101) f ( x ) = − f ( x ). Now J ( τ ) = I ( τ ) and we see from the definitions of fractalproperties (oscillatory and curve dimensions and Minkowski contents) of anoscillatory integral, that they are invariant to complex conjugation of I .We use the asymptotic expansion of the integral I from (1), I ( τ ) ∼ e iτf (0) ∞ (cid:88) j =1 C j · τ − j/s , as τ → ∞ , where C j ∈ C , from [13, Proposition 3 on page 334], and it holds that C (cid:54) = 0. From the same reference, it follows that each constant C j dependson only finitely many derivatives of f and φ at 0. We write(7) I ( τ ) = e iτf (0) P ( τ ) , where the function P ( τ ) ∼ ∞ (cid:80) j =1 C j · τ − j/s , as τ → ∞ .First we show that the function I is of class C ∞ ( R ), using derivationunder the integral sign. By taking the derivative of (1), we get I (cid:48) ( τ ) = i (cid:90) R n e iτf ( x ) φ ( x ) dx, SCILLATORY INTEGRALS AND FRACTAL DIMENSION 13 where φ ( x ) = f ( x ) φ ( x ). Inductively, we see that I ( m ) ( τ ) = i m (cid:90) R n e iτf ( x ) φ m ( x ) dx, for all m ∈ N , where φ m ( x ) = [ f ( x )] m φ ( x ). Notice that for every m ∈ N , the function I ( m ) is equal to the constant i m multiplying the oscillatory integral of type(1), with the phase f and the amplitude φ m . Further, using the asymptoticexpansion of this integral, we get(8) I ( m ) ( τ ) = i m e iτf (0) P m ( τ ) , for all m ∈ N , where the function P m possesses an asymptotic expansion in the same as-ymptotic sequence as P .Now we want to prove that the function P is of class C ∞ ( R ), and thatits derivative of any order possesses an asymptotic expansion in the sameasymptotic sequence as P , that is, P ( m ) ( τ ) ∼ ∞ (cid:80) j =1 C ( m ) j · τ − j/s , as τ → ∞ ,where C ( m ) j ∈ C . Notice that from (7) and the fact that I ∈ C ∞ immediatelyfollows that P ∈ C ∞ .By taking the derivative of (7), we get I (cid:48) ( τ ) = if (0) e iτf (0) P ( τ ) + e iτf (0) P (cid:48) ( τ ) . Respecting (8) and dividing every term by e iτf (0) , we get the expression P (cid:48) ( τ ) = i ( P ( τ ) − f (0) P ( τ )) , using [3, p. 14], it shows that P (cid:48) possesses an asymptotic expansion in thesame asymptotic sequence as P . It follows by induction, that for all m ∈ N , the function P ( m ) also possesses an asymptotic expansion in the sameasymptotic sequence as P .Notice that the exponents of the monomials of the asymptotic sequenceare integer multiples of a common real number − /s . Hence, it follows fromthe clasical proof; see [3, p. 21], that the asymptotic expansion of P ( m ) isgiven by m times differentiating the asymptotic expansion of P , term byterm.Now define a j = Re C j and b j = Im C j for all j ∈ N . Also, definefunctions A ( τ ) = Re P ( τ ) and B ( τ ) = Im P ( τ ). Respecting (2) we get X ( τ ) = cos( τ f (0)) A ( τ ) − sin( τ f (0)) B ( τ ) , (9) Y ( τ ) = sin( τ f (0)) A ( τ ) + cos( τ f (0)) B ( τ ) , (10)where A ( τ ) ∼ ∞ (cid:88) j =1 a j · τ − j/s , as τ → ∞ , (11) B ( τ ) ∼ ∞ (cid:88) j =1 b j · τ − j/s , as τ → ∞ . (12) Notice that from P ( τ ) = A ( τ ) + iB ( τ ) follows that functions A and B are of class C ∞ ( R ) and that A ( m ) and B ( m ) , m ∈ N , possess asymptoticexpansions given by m times differentiating the asymptotic expansions of A and B , term by term, respectively.For the oscillatory dimension we will provide the proof for Y . For thefunction X the proof is analogous. Using the substitution t = 1 /τ , to deter-mine the oscillatory dimension of Y , we further investigate the asymptoticexpansion of the function y , defined by y ( t ) = Y (1 /t ), near the origin, y ( t ) = sin( f (0) /t ) a ( t ) + cos( f (0) /t ) b ( t ) , where a ( t ) = A (cid:0) t − (cid:1) and b ( t ) = B (cid:0) t − (cid:1) . Now a ( t ) ∼ ∞ (cid:88) j =1 a j · t j/s , as t → + , (13) b ( t ) ∼ ∞ (cid:88) j =1 b j · t j/s , as t → + . (14)Notice that both functions a and b are of class C ∞ ( R + ) and that both a ( m ) and b ( m ) , for all m ∈ N , possess asymptotic expansions, near the origin,given by d m dt m (cid:2) A (cid:0) t − (cid:1)(cid:3) and d m dt m (cid:2) B (cid:0) t − (cid:1)(cid:3) , respectively. Indeed, this m -thderivatives are finite linear combinations of products given by A ( k ) (cid:0) t − (cid:1) or B ( k ) (cid:0) t − (cid:1) , multiplied by a negative power of t , where k ≤ m . A linearcombination of asymptotic expansions is again an asymptotic expansion; see[3, p. 14].Finally, we define functions p ( t ) = (cid:112) a ( t ) + b ( t ) and ψ : R → [0 , π )such that cos ψ ( t ) = a ( t ) p ( t ) , sin ψ ( t ) = b ( t ) p ( t ) . Exploiting trigonometric addition formulas we get the expression y ( t ) = p ( t ) sin( q ( t )) , where q ( t ) = f (0) · t − + ψ ( t ) . The function p is of class C ∞ ( R + ), and q is also C ∞ ( R + ), as ψ is C ∞ ( R + )by the definition, for sufficiently small t . For the derivative of q , we get(15) q (cid:48) ( t ) = − f (0) · t − + ψ (cid:48) ( t ) , where ψ ( t ) = arctan b ( t ) a ( t ) . Because b ( t ) /a ( t ) → const as t → ∞ , and as arctan is an analytic function,then for all m ∈ N , ψ ( m ) ( t ) and q ( m ) possess an asymptotic expansion.Using the same principle, we can prove that for all m ∈ N , p ( m ) possessesan asymptotic expansion. SCILLATORY INTEGRALS AND FRACTAL DIMENSION 15
The asymptotic representation of the function p is easily determined, p ( t ) = (cid:113)(cid:0) a t /s + O (cid:0) t /s (cid:1)(cid:1) + (cid:0) b t /s + O (cid:0) t /s (cid:1)(cid:1) (16) = t /s (cid:113) a + b (cid:16) O (cid:16) t /s (cid:17)(cid:17) ∼ t /s (cid:113) a + b , (17)as t →
0, and the asymptotic representation of derivative of any order of p is given by differentiating that many times the asymptotic representation of p . For the function q , as ψ is bounded, it follows that q ( t ) ∼ f (0) · t − , as t →
0, and for all m ∈ N , q ( m ) ∼ f (0) · d m dt m (cid:2) t − (cid:3) , as t → S ( t ) = sin t , and constants T = π , α = 1 /s and β = 1. Notice that all of the assumptions of thattheorem are satisfied. Let Γ y be the graph of the function y . We concludethat dim B Γ y = dim osc Y = 2 − α +1 β +1 = s − s and that Γ y is Minkowskinondegenerate.In order to compute the curve dimension, we want to investigate theoscillatory integral I in polar coordinates. We first define the real function G ( τ ) = | I ( τ ) | = | P ( τ ) | = (cid:112) X ( τ ) + Y ( τ ) = (cid:112) A ( τ ) + B ( τ ), hence itfollows that G is of class C ∞ . Using asymptotic expansions for A and B ,we get G ( τ ) ∼ ∞ (cid:88) j =1 c j · τ − j/s , as τ → ∞ , where c j = (cid:113) a j + b j = | C j | ∈ R , for all j ∈ N . Next, we define thecontinuous function ϕ : [ τ , ∞ ) → R , τ >
0, bytan ϕ ( τ ) = Y ( τ ) X ( τ ) , where X ( τ ) (cid:54) = 0, and extend it by continuity. The zero set of X is discretebecause of the asymptotic expansion of X (cid:48) ( τ ). Using trigonometric additionformulas we calculatetan ϕ ( τ ) = sin( τ f (0) + Ψ( τ )) G ( τ )cos( τ f (0) + Ψ( τ )) G ( τ ) = tan( τ f (0) + Ψ( τ )) , where Ψ : [ τ , ∞ ) → [0 , π ) is such thatcos Ψ( τ ) = A ( τ ) G ( τ ) , sin Ψ( τ ) = B ( τ ) G ( τ ) . We compute the expressionΨ (cid:48) ( τ ) = A ( τ ) B (cid:48) ( τ ) − B ( τ ) A (cid:48) ( τ ) A ( τ ) + B ( τ ) = K · τ − − /s (cid:16) O (cid:16) τ − /s (cid:17)(cid:17) , as τ → ∞ , where the constant K = a b − a b s (cid:0) a + b (cid:1) , so ϕ (cid:48) ( τ ) = f (0) + Ψ (cid:48) ( τ ) ∼ f (0) + K · τ − − /s , as τ → ∞ . From the expression for Ψ (cid:48) ( τ ) it follows that Ψ (cid:48) is of class C ∞ . As Ψ is a continuous function for sufficiently large τ , it follows that Ψ is of class C ∞ and it holds ϕ ( τ ) = τ f (0) + Ψ( τ ), hence ϕ is also of class C ∞ . Analogously as before, functions G ( m ) , ϕ ( m ) and Ψ ( m ) possess asymptotic expansions for all m ∈ N . As f (0) >
0, we can take τ > ϕ (cid:48) ( τ ) >
0, for every τ ∈ [ τ , ∞ ). Asnow ϕ : [ τ , ∞ ) → [ ϕ , ∞ ), where ϕ = ϕ ( τ ), is of class C ∞ and a strictlyincreasing bijection, so is its inverse function τ : [ ϕ , ∞ ) → [ τ , ∞ ).Now the radius function r : [ ϕ , ∞ ) → [0 , ∞ ), defined by r ( ϕ ) = G ( τ ( ϕ )),is of class C ∞ . We want to determine asymptotic representations of twoderivatives of r ( ϕ ), as ϕ → ∞ . From before, we know that G ( τ ) ∼ c · τ − /s , G (cid:48) ( τ ) ∼ − c s · τ − − /s and G (cid:48)(cid:48) ( τ ) ∼ c s (cid:0) s (cid:1) · τ − − /s , as τ → ∞ . Wecompute r (cid:48) ( ϕ ) = G (cid:48) ( τ ( ϕ )) τ (cid:48) ( ϕ ) = G (cid:48) ( τ ( ϕ )) ϕ (cid:48) ( τ ( ϕ )) ,r (cid:48)(cid:48) ( ϕ ) = G (cid:48)(cid:48) ( τ ( ϕ ))( τ (cid:48) ( ϕ )) + G (cid:48) ( τ ( ϕ )) τ (cid:48)(cid:48) ( ϕ )= G (cid:48)(cid:48) ( τ ( ϕ ))( ϕ (cid:48) ( τ ( ϕ ))) − G (cid:48) ( τ ( ϕ )) ϕ (cid:48)(cid:48) ( τ ( ϕ ))[ ϕ (cid:48) ( τ ( ϕ ))] , since τ (cid:48) ( ϕ ) = [ ϕ (cid:48) ( τ ( ϕ ))] − and τ (cid:48)(cid:48) ( ϕ ) = − ϕ (cid:48)(cid:48) ( τ ( ϕ )) / [ ϕ (cid:48) ( τ ( ϕ ))] .As Ψ is a bounded function, it follows that ϕ ( τ ) ∼ τ f (0), as τ → ∞ .It is easy to see that the inverse τ ( ϕ ) ∼ ϕ/f (0), as ϕ → ∞ . Notice that ϕ (cid:48) ( τ ) ∼ f (0) and ϕ (cid:48)(cid:48) ( τ ) ∼ K (cid:0) − − s (cid:1) · τ − − /s , as τ → ∞ . Finally, noticethat τ ( ϕ ) → ∞ , as ϕ → ∞ . Hence, we can compute r ( ϕ ) ∼ c · ( ϕ/f (0)) − /s = c f (0) /s ϕ − /s ,r (cid:48) ( ϕ ) ∼ − c s · ( ϕ/f (0)) − − /s f (0) = − c s f (0) /s · ϕ − − /s ,r (cid:48)(cid:48) ( ϕ ) ∼ c s (cid:0) s (cid:1) · ( ϕ/f (0)) − − /s ( f (0)) + c s · ( ϕ/f (0)) − − /s K (cid:0) − − s (cid:1) · ( ϕ/f (0)) − − /s ( f (0)) ∼ c s (cid:18) s (cid:19) f (0) /s · ϕ − − /s , as ϕ → ∞ . Notice, as c > r (cid:48) ( ϕ ) <
0, for ϕ sufficiently large, so wecan take τ > r is a strictly decreasing function.Finally, we use Theorem 5 from Section 5, taking α = 1 /s . Function r satisfies the assumptions of this theorem. We calculate the constant m from(23), below, to be equal to f (0) /s | C | , and | r (cid:48)(cid:48) ( ϕ ) ϕ α | →
0, as ϕ → ∞ , soit is uniformly bounded as a function of ϕ on its domain [ ϕ , ∞ ). As all ofthe assumptions of that theorem are satisfied, we conclude that the curvedimension of I is d := 2 / (1 + α ) = 2 s/ ( s + 1), the curve Γ is Minkowskimeasurable and its d -dimensional Minkowski content is given by (4). (cid:3) SCILLATORY INTEGRALS AND FRACTAL DIMENSION 17
Proof of Theorem 2.
Using [2, Theorem 6.5] we conclude that the oscillationindex of the critical point of f equals to its remoteness β . Then using a k,γ := a k,γ ( φ ) and rewriting (3) we get the asymptotic expansion(18) I ( τ ) ∼ e iτf (0) a ,β τ β log τ + a ,β τ β + (cid:88) α<β ( a ,α τ α log τ + a ,α τ α ) as τ → ∞ , where α runs through a finite set of arithmetic progressions,hence there exists ε such that | β − α | > ε for all such α .Without loss of generality, we can assume that we work in superadapted ,hence adapted coordinates; see [5, Section 7.]. We now have to establish,for cases ( i ) and ( ii ), if the first coefficient a ,β is vanishing or not.For case ( i ), as the multiplicity of the remoteness is equal to 0 and thedimension n = 2, we conclude that the open face of the Newton diagramof the phase f that contains the center of the boundary of the associatedNewton polyhedron is an edge. If the Newton polyhedron is remote, thatis β > −
1, we are in the Case 1 or 3 from [5, Theorem 1.2], from which itfollows that a ,β = 0. From the definition of the oscillation index it followsthat a ,β (cid:54) = 0. If β = −
1, it follows from [5, lemma 1.0] that the criticalpoint of f at the origin in nondegenerate. Now, from [2, Theorem 6.2] itfollows that a ,β (cid:54) = 0 and a ,β = 0.The rest of the proof now basically follows the proof of Theorem 1. Mi-nor differences arise regarding treatment of more complicated asymptoticexpansion (18), which has terms having a logarithm function.For case ( ii ), the multiplicity of the remoteness is equal to 1, hence thecenter of the boundary of the associated Newton polyhedron is a vertex. As β > −
1, we are in the Case 2 from [5, Theorem 1.2], hence from [5, Comment2.] it follows that a ,β (cid:54) = 0. Now the first term in the asymptotic expansionhas a logarithm inside. Like in the case ( i ), the proof of Theorem 1 isonce more adapted concerning log-terms in asymptotic expansions. Furtherdifferences arise in the final steps of the proof, when applying Theorems 4and 5, for oscillatory and curve dimensions, respectively.We first consider the proof for the oscillatory dimension. It is easy tosee that here, contrary to the proof of Theorem 1, it holds p ( t ) ∼ const · t − α log( t − ), as t →
0. It follows p (cid:48) ( t ) ∼ const · t − α − log( t − ), as t →
0. Soinstead of Theorem 4 from Section 5, which can not be applied here, we useTheorem 6. For the proof for the curve dimension, instead of using Theorem5 on the curve radius function r = r ( ϕ ) (see the proof of Theorem 1), weuse directly Theorem 10. (cid:3) Proof of Theorem 3.
Using [2, Theorem 6.4] we conclude that the oscillationindex of the critical point of f equals to the remoteness β . Using (3) we get the asymptotic expansion(19) I ( τ ) ∼ e iτf (0) n − (cid:88) k =0 a k,β ( φ ) τ β log k τ + (cid:88) α<β n − (cid:88) k =0 a k,α ( φ ) τ α log k τ as τ → ∞ , where α runs through a finite set of arithmetic progressions.The rest of the proof is analogous to the proof of Theorem 2, in both cases.Notice that here the asymptotic scale is involving terms consisting of τ to anegative rational power multiplied by a logarithm of τ to the power k . (cid:3) Fractal properties of chirps and spirals related tooscillatory integrals
In order to compute curve and oscillatory dimensions of oscillatory in-tegrals and related Minkowski contents, we use theorems presented in thissection. Theorems 4 and 5, cited below, were used before in different setting,related to fractal analysis of differential equations, Fresnel integrals and dy-namical systems; see [7], [6] and [16]. Here, they are used in the proofs ofTheorems 1, 2 and 3. Also, for the proofs of Theorems 2 and 3, Theorems 4and 5 had to be modified, as the original versions fail to take into accountthe power-log asymptotic of the leading term in the asymptotic expansionof related oscillatory integrals. This modified variants, Theorems 6 and 10below, are proved throughout the rest of this section.
Theorem . Let y ( x ) = p ( x ) S ( q ( x )) , where x ∈ I =(0 , c ] and c > . Let the functions p ( x ) , q ( x ) and S ( t ) satisfy the followingassumptions: (20) p ∈ C ( ¯ I ) ∩ C ( I ) , q ∈ C ( I ) , S ∈ C ( R ) . The function S ( t ) is assumed to be a T -periodic real function defined on R such that (21) (cid:26) S ( a ) = S ( a + T ) = 0 for some a ∈ R , S ( t ) (cid:54) = 0 for all t ∈ ( a, a + T ) ∪ ( a + T, a + 2 T ) ,where T is a positive real number and S ( t ) alternately changes a sign onintervals ( a + ( k − T, a + kT ) , for k ∈ N . Without loss of generality, wetake a = 0 . Let us suppose that < α ≤ β and: (22) p ( x ) (cid:39) x α as x → , q ( x ) (cid:39) x − β as x → . Let Γ y be the graph of the function y . Then dim B Γ y = 2 − ( α + 1) / ( β + 1) and Γ y is Minkowski nondegenerate.Theorem . Assume that ϕ > and r : [ ϕ , ∞ ) → (0 , ∞ ) is a decreasing C function converging to zero as ϕ → ∞ . Let thelimit (23) m := lim ϕ →∞ r (cid:48) ( ϕ )( ϕ − α ) (cid:48) SCILLATORY INTEGRALS AND FRACTAL DIMENSION 19 exist, where α ∈ (0 , . Assume that | r (cid:48)(cid:48) ( ϕ ) ϕ α | is uniformly bounded asa function of ϕ . Let Γ be the graph of the spiral ρ = r ( ϕ ) and define d := 2 / (1 + α ) . Then dim B Γ = d , the spiral is Minkowski measurable, andmoreover, (24) M d (Γ) = m d π ( πα ) − α/ (1+ α ) α − α . Remark . Theorem 5 is the simplified but equivalent form of the result firstintroduced in [16].
Theorem . (Box dimension and Minkowski degeneracy of the graphof a logarithmic ( α, -chirp-like function) Let y ( x ) = p ( x ) sin( q ( x )) , x ∈ I = (0 , c ] , c > . Let the functions p ( x ) and q ( x ) satisfy the followingassumptions: (25) p ∈ C ( ¯ I ) ∩ C ( I ) , q ∈ C ( I ) . Let us suppose that < α ≤ , l ∈ N and: (26) p ( x ) (cid:39) x α (cid:2) log( x − ) (cid:3) l as x → , (27) q ( x ) (cid:39) x − as x → . Let Γ y be the graph of the function y . Then dim B Γ y = d , where d =2 − ( α + 1) / , and Γ y is Minkowski degenerate, having M d (Γ y ) = ∞ .Remark . Theorem 6 is a modified variant of Theorem 4, by setting S ( x ) =sin x , T = π and β = 1 in the original theorem, and adapting condition(22) by introduction of log-term asymptotics. The same applies also forPropositions 3 and 4, below.The proof of Theorem 6 uses Theorem 7, below, which is a modifiedvariant of [11, Theorem 2.1.], and two propositions concerning the propertiesof functions p and q , which are also modified variants of [7, Proposition 1 and2], below. We also need [11, Definition 2.1.], stating that for some ε >
0, wesay that a function k = k ( ε ) is an index function on (0 , ε ] if k : (0 , ε ] → N , k ( ε ) is nonincreasing and lim ε → k ( ε ) = ∞ . Theorem . Let y ∈ C ((0 , T ]) bea bounded function on (0 , T ] . Let s ∈ [1 , be a real number, let l ∈ N andlet ( a n ) be a decreasing sequence of consecutive zeros of y ( x ) in (0 , T ] suchthat a n → when n → ∞ and let there exist constants c , c , ε such thatfor all ε ∈ (0 , ε ) we have: (28) c ε − s (cid:2) log( ε − ) (cid:3) l ≤ (cid:88) n ≥ k ( ε ) max x ∈ [ a n +1 ,a n ] | y ( x ) | ( a n − a n +1 ) , (29) a k ( ε ) sup x ∈ (0 ,a k ( ε ) ] | y ( x ) | + ε (cid:90) a a k ( ε ) | y (cid:48) ( x ) | dx ≤ c ε − s (cid:2) log( ε − ) (cid:3) l , where k ( ε ) is an index function on (0 , ε ] such that | a n − a n +1 | ≤ ε for all n ≥ k ( ε ) and ε ∈ (0 , ε ) . Let G ( y ) be the graph of the function y . Then dim B ( G ( y )) = s and G ( y ) is Minkowski degenerate, having M s ( G ( y )) = ∞ .Proof. Let G ε ( y ) be the ε -neighbourhood of the graph G ( y ) of the function y . From [11, Lemma 2.1.] it follows that | G ε ( y ) | ≥ c ε − s (cid:2) log( ε − ) (cid:3) l , andfrom [11, Lemma 2.2.] it follows that | G ε ( y ) | ≤ c (cid:104) ε + c ε − s (cid:2) log( ε − ) (cid:3) l (cid:105) ,where c >
0. From the definitions of M ∗ s ( G ( y )) and M s ∗ ( G ( y )) it followsthat M ∗ s ( G ( y )) ≥ M s ∗ ( G ( y )) ≥ lim inf ε → c ε − s (cid:2) log( ε − ) (cid:3) l ε − s = + ∞ , and that M s (cid:48) ∗ ( G ( y )) ≤ M ∗ s (cid:48) ( G ( y )) ≤ lim inf ε → c (cid:104) ε + c ε − s (cid:2) log( ε − ) (cid:3) l (cid:105) ε − s (cid:48) = 0holds for all s (cid:48) > s , hence the theorem is proved. (cid:3) Proposition . Assume that thefunctions p ( x ) and q ( x ) satisfy conditions (25) , (26) and (27) . Then thereexist δ > , l ∈ N and positive constants C and C such that: C x α (cid:2) log( x − ) (cid:3) l ≤ p ( x ) ≤ C x α (cid:2) log( x − ) (cid:3) l ,C x α − (cid:2) log( x − ) (cid:3) l ≤ p (cid:48) ( x ) ≤ C x α − (cid:2) log( x − ) (cid:3) l ,C x − ≤ q ( x ) ≤ C x − ,C x − ≤ − q (cid:48) ( x ) ≤ C x − , for all x ∈ (0 , δ ] . Furthermore, there exists the inverse function q − of thefunction q defined on [ m , ∞ ) , where m = q ( δ ) , and it holds: q − ( t ) (cid:39) t − as t → ∞ ,C t − ( t − s ) ≤ q − ( s ) − q − ( t ) ≤ C s − ( t − s ) , m ≤ s < t. Proposition . For any function q ( x ) with properties (25) and (27) , we have: (i) Let a k = q − ( kπ ) and s k = q − ( t + kπ ) , k ∈ N , where t ∈ (0 , π ) isarbitrary. Then there exist k ∈ N and c > such that a k ∈ (0 , δ ] , y ( a k ) = 0 , s k ∈ ( a k +1 , a k ) for all k ≥ k , a k (cid:38) as k → ∞ , a k (cid:39) k − as k → ∞ , and max x ∈ [ a k +1 ,a k ] | y ( x ) | ≥ c ( k + 1) − α [log ( k + 1)] l for all k ≥ k , c > , where y ( x ) = p ( x ) sin( q ( x )) and the function p ( x ) and l ∈ N satisfy(26). SCILLATORY INTEGRALS AND FRACTAL DIMENSION 21 (ii)
There exists ε > and a function k : (0 , ε ) → N such that (30) 1 π (cid:18) επC (cid:19) − ≤ k ( ε ) ≤ π (cid:18) επC (cid:19) − . In particular, C π ( k + 1) − ≤ a k − a k +1 ≤ ε, for all k ≥ k ( ε ) and ε ∈ (0 , ε ) . Proofs of Propositions 3 and 4 are analogous as in [7].
Proof of Theorem 6.
We have to check that assumptions (28) and (29) aresatisfied. By Proposition 4 we have (cid:88) k ≥ k ( ε ) max x ∈ [ a k +1 ,a k ] | y ( x ) | ( a k − a k +1 ) ≥ c C π ∞ (cid:88) k = k ( ε ) ( k + 1) − α − [log( k + 1)] l ≥ c ∞ (cid:88) k = k ( ε )+1 k − α − [log k ] l = ca, where the series a = (cid:80) ∞ k = k ( ε )+1 k − α − [log k ] l is convergent. Then, using theintegral test for convergence and (30), we obtain that ca ≥ (cid:90) ∞ k = k ( ε )+1 k − α − [log k ] l ≥ c ( 1 k ( ε ) + 1 ) α +1 [log( k ( ε ) + 1)] l ≥ c k ( ε ) ) α +1 [log k ( ε )] l ≥ c ε − ( − α +12 ) (cid:2) log (cid:0) ε − (cid:1)(cid:3) l , for all ε ∈ (0 , ε ). Using Proposition 3 it follows that | y (cid:48) ( x ) | = | p (cid:48) ( x ) sin( q ( x )) + p ( x ) q (cid:48) ( x ) cos( q ( x )) | ≤ c x α − (cid:2) log (cid:0) x − (cid:1)(cid:3) l ≤ c x α − , which holds near x = 0 + . By Proposition 4 we conclude that a k ( ε ) sup x ∈ (0 ,a k ( ε ) ] | y ( x ) | + ε (cid:90) a k a k ( ε ) | y (cid:48) ( x ) | dx ≤ c ε α +12 (cid:2) log (cid:0) ε − (cid:1)(cid:3) l + εc [ a α − k + a α − k ( ε ) ] ≤ c ε − ( − α +12 ) (cid:2) log (cid:0) ε − (cid:1)(cid:3) l , for all ε ∈ (0 , ε ).Finally, we apply Theorem 7, where s = 2 − α +12 . (cid:3) The last part of this section is devoted to proving the modified variantof Theorem 5. More precisely, we will prove the modified variant of theoriginal result, [16, Theorem 5]. To prove this variant, Theorem 10 below,we proceed our presentation as in [16], by first stating and proving wherenecessary, some auxiliary definitions and results.
We define a spiral in the plane as the graph Γ of a function r = f ( ϕ ), ϕ ≥ ϕ , in polar coordinates, where f : [ ϕ , ∞ ) → (0 , ∞ ) is such that f ( ϕ ) → ϕ → ∞ , f is radially decreasing (ie, for any fixed ϕ ≥ ϕ the function N (cid:51) k (cid:55)→ f ( ϕ + 2 kπ ) is decreasing)(31)Let Γ be a spiral defined by r = f ( ϕ ), ϕ ≥ ϕ . We denote a subset of thespiral Γ corresponding to angles in the interval ( ϕ , ϕ ) by Γ( ϕ , ϕ ), moreprecisely,(32) Γ( ϕ , ϕ ) := { ( r, ϕ ) ∈ Γ : ϕ ∈ ( ϕ , ϕ ) } . Let A be a bounded set in R N , and let the radial distance function d rad ( x, A ), be defined as the Euclidean distance from x to the set A ∩{ tx : t ≥ } , provided the intersection is nonempty, and ∞ otherwise. Now theradial ε -neighbourhood around A is defined as the set A ε,rad := { y ∈ R N : d rad ( y, A ) < ε } .Using radial ε -neighbourhood we define radial s-dimensional lower andupper Minkowski content of set A , analogously as in Section 1.1, denotedby M s ∗ ( A, rad ) and M ∗ s ( A, rad ), respectively. Also, analogously we define radial lower and radial upper box dimension of A , denoted by dim B ( A, rad )and dim B ( A, rad ), respectively. If both quantities coincide, the commonvalue is denoted by dim B ( A, rad ), and we call it radial box dimension of A .For a general definition of directional box dimensions in R , see Tricot [14,pp. 248–249]. Since A ε,rad ⊆ A ε , it is clear that(33) dim B ( A, rad ) ≤ dim B A, dim B ( A, rad ) ≤ dim B A. We define (radial) ε - nucleus of the spiral Γ as the radial ε -neighbourhoodaround Γ( ϕ ( ε ) , ∞ ) ⊂ Γ, that is,(34) N (Γ , ε ) := Γ( ϕ ( ε ) , ∞ ) ε,rad , where by ϕ ( ε ) we denote the smallest angle such that for all ψ ≥ ϕ ( ε ) wehave f ( ψ ) − f ( ψ + 2 π ) ≤ ε , more precisely,(35) ϕ ( ε ) := inf { ϕ ≥ ϕ : ∀ ψ ≥ ϕ, f ( ψ ) − f ( ψ + 2 π ) ≤ ε } . The set T (Γ , ε ) obtained as the radial ε -neighbourhood around the arcΓ( ϕ , ϕ ( ε )), that is,(36) T (Γ , ε ) := Γ( ϕ , ϕ ( ε )) ε,rad , is called (radial) ε - tail of the spiral Γ. The notions of nucleus and tail of aspiral are introduced by Tricot [14, pp. 121, 122].We consider lower nucleus and lower tail s -dimensional Minkowski con-tents of Γ defined by(37) M s ∗ (Γ , n ) := lim inf ε → | N (Γ , ε ) | ε − s , M s ∗ (Γ , t ) := lim inf ε → | T (Γ , ε ) | ε − s SCILLATORY INTEGRALS AND FRACTAL DIMENSION 23 respectively, for s ≥
0. Analogously for the upper nucleus and upper tailMinkowski contents. It is clear that(38) M ∗ s (Γ , rad ) ≤ M ∗ s (Γ , n ) + M ∗ s (Γ , t ) . Indeed, we can express radial ε -neighbourhood around Γ as Γ ε,rad = N (Γ , ε ) ∪ T (Γ , ε ) ∪ S ( ε ), where S ( ε ) := { ( r, ϕ ) ∈ Γ ε,rad : ϕ = ϕ ( ε ) } is of the 2-dimensional Lebesgue measure zero. Hence M ∗ s (Γ , rad ) ≤ lim sup ε → | N (Γ , ε ) | ε − s + lim sup ε → | T (Γ , ε ) | ε − s . First we prove the modified variant of [16, Theorem 1].
Theorem . Let f : [ ϕ , ∞ ) → (0 , ∞ ) , where ϕ > e , be a measurable, ra-dially decreasing function, see (31). Let α ∈ (0 , and l ∈ N such that forsome positive numbers m and m we have (39) m ϕ − α [log ϕ ] l ≤ f ( ϕ ) ≤ m ϕ − α [log ϕ ] l for all ϕ ≥ ϕ > . Assume that there exist positive constants a and a suchthat for all ϕ ≥ ϕ , (40) a ϕ − α − [log ϕ ] l ≤ f ( ϕ ) − f ( ϕ + 2 π ) ≤ a ϕ − α − [log ϕ ] l . Let Γ be the graph of r = f ( ϕ ) in polar coordinates. Then d := dim B (Γ , rad ) = α , (41) M ∗ d (Γ , rad ) = + ∞ . (42) Proof.
We first obtain the upper bound of the area of the ε -nucleus of Γ.Note that inequality f ( ϕ ) − f ( ϕ +2 π ) > ε is satisfied when a ϕ − α − [log ϕ ] l >a ϕ − α − > ε , that is, for ϕ < ϕ ( ε ), where ϕ ( ε ) := (cid:16) εa (cid:17) − / (1+ α ) . Fromthe definition of ϕ ( ε ), see (35), we have(43) ϕ ( ε ) ≥ ϕ ( ε ) , therefore,(44) | N (Γ , ε ) | ≤ π ( sup [ ϕ ( ε ) ,ϕ ( ε )+2 π ] f + ε ) ≤ π (cid:18) m ϕ ( ε ) − α (cid:104) log ϕ ( ε ) (cid:105) l + ε (cid:19) . We see that(45) | N (Γ , ε ) | ≤ c · ε α/ (1+ α ) [log ε ] l , where c > ε -tail of Γ from above. The inequality f ( ϕ ) − f ( ϕ + 2 π ) < ε is satisfied when a ϕ − α − [log ϕ ] l < ε . Hence, f ( ϕ ) − f ( ϕ + 2 π ) < ε is satisfied for ϕ > ϕ ( ε ), where(46) ϕ ( ε ) := (cid:18) εa (cid:19) − / (1+ α − δ ) , for δ = δ ( ε ) := inf δ> { δ : [log ϕ ] l < ϕ δ , ∀ ϕ ≥ ϕ ( ε ) } . Notice that a ϕ − α − [log ϕ ] l ≤ a ϕ − ( α − δ ) − , for all ϕ ≥ ϕ ( ε ). Therefore ϕ ( ε ) ≤ ϕ ( ε ), and from this wehave that | T (Γ , ε ) | ≤ (cid:90) ϕ ( ε ) ϕ [( f ( ϕ ) + ε ) − ( f ( ϕ ) − ε ) ] dϕ = 2 ε (cid:90) ϕ ( ε ) ϕ f ( ϕ ) dϕ ≤ εm (cid:90) ϕ ( ε ) ϕ ϕ − α [log ϕ ] l dϕ ≤ εm (cid:90) ϕ ( ε ) ϕ ϕ − ( α − δ ) dϕ = 2 εm − α ( ϕ ( ε ) − ( α − δ ) − ϕ − ( α − δ )1 ) ≤ c · ε α − δ ) / (1+ α − δ ) , where c >
0. Notice that δ → ε → d := 2 / (1 + α ) we have that, see (38),(47) M ∗ d (Γ , rad ) ≤ M ∗ d (Γ , n ) + M ∗ d (Γ , t ) = + ∞ + ∞ = + ∞ . For every d (cid:48) > d it holds that M ∗ d (cid:48) (Γ , n ) = 0, and we can take ε > δ > M ∗ d (cid:48) (Γ , t ) =0. Hence, we conclude that dim B (Γ , rad ) ≤ d .To obtain a lower bound of the area of ε -nucleus of Γ, we show that(48) N (Γ , ε ) ⊃ B r (0) , r := inf ϕ ∈ [ ϕ ( ε ) ,ϕ ( ε )+2 π ] f ( ϕ ) , analogously as in the proof of [16, Theorem 1]. Using (48) and (39) weobtain(49) | N (Γ , ε ) | ≥ πr ≥ π (cid:16) m ( ϕ ( ε ) + 2 π ) − α [log( ϕ ( ε ))] l (cid:17) , hence,(50) | N (Γ , ε ) | ≥ c · ε α/ (1+ α − δ ) [log ε ] l , where c > | T (Γ , ε ) | ≥ ε (cid:90) ϕ ( ε ) ϕ f ( ϕ ) dϕ ≥ c · ε α/ (1+ α ) , where c >
0, provided ε is sufficiently small.As ϕ ( ε ) ≤ ϕ ( ε ), we conclude that(52) M d ∗ (Γ , rad ) ≥ lim inf ε → c · ε α/ (1+ α − δ ) [log ε ] l + c · ε α/ (1+ α ) ε − d = c . For every d (cid:48) < d , we can take ε > δ > M d (cid:48) ∗ (Γ , rad ) = + ∞ . Hence, we concludethat dim B (Γ , rad ) ≥ d . (cid:3) SCILLATORY INTEGRALS AND FRACTAL DIMENSION 25
The following theorem is a marginally modified variant of [16, Theorem4]. The only difference is in adding the log term [log ε ] l . Theorem . Let Γ be a spiral of focus type defined by r = f ( ϕ ) , f : [ ϕ , ∞ ) → (0 , ∞ ) , such that f ( ϕ ) is decreasing, and f ( ϕ ) → . Let there exist ε suchthat the functional inequality (53) f (cid:18) ϕ + εf ( ϕ ) + εf ( ϕ ) − ε (cid:19) > f ( ϕ ) − ε. holds for all ε ∈ (0 , ε ) and ϕ ∈ ( ϕ , ϕ ( ε )) , where ϕ ( ε ) is defined by (35).Assume also that there exist positive constants C , C and q < such that C ε q ≤ f ( ϕ ( ε )) ≤ C ε − d [log ε ] l , where d := dim B (Γ , rad ) and l ∈ N . Then (54) dim B Γ = dim B (Γ , rad ) . We omit the proof of Theorem 9, as it is almost completely analogousto the proof of [16, Theorem 4]. Only one small difference occurs in thetreatment of the log term in the condition f ( ϕ ( ε )) ≤ C ε − d [log ε ] l .The following excision property of Minkowski contents will enable us tohandle the condition for ϕ to be sufficiently large in Theorem 10. Wecompletely omit the proof, as it is already proved in [16]. Lemma . (Excision property for simple smooth curves) Let Γ be a simplesmooth curve in R , that is, Γ is the graph of continuously differentiableinjection h : [ ϕ , ∞ ) → R . Assume that dim B Γ > . Let ϕ > ϕ be givenand Γ := h ( ϕ , ∞ ) . Then d := dim B Γ = dim B Γ , d := dim B Γ = dim B Γ , (55) M d ∗ (Γ ) = M d ∗ (Γ) , M ∗ d (Γ ) = M ∗ d (Γ) . (56) Analogous claim holds for radial box dimensions and radial Minkowski con-tents: if dim B (Γ , rad ) > , then δ := dim B (Γ , rad ) = dim B (Γ , rad ) , δ := dim B (Γ , rad ) = dim B (Γ , rad ) , M δ ∗ (Γ , rad ) = M δ ∗ (Γ , rad ) , M ∗ δ (Γ , rad ) = M ∗ δ (Γ , rad ) . In particular, the conclusions hold for smooth spirals r = f ( ϕ ) , where f ( ϕ ) is a decreasing function tending to as ϕ → ∞ . Finally, here is the modified variant of Theorem 5.
Theorem . Assume in addition to the assumptions of Theorem 8 that thefunction f is decreasing, of class C , and there exist positive constants M and M such that for all ϕ ≥ ϕ , (57) M ϕ − α − [log ϕ ] l ≤ | f (cid:48) ( ϕ ) | ≤ M ϕ − α − [log ϕ ] l . Then (58) dim B Γ = dim B (Γ , rad ) = d, and (59) M ∗ d (Γ) = M ∗ d (Γ , rad ) = + ∞ , where d := α .Proof. (a) From the excision result, see Lemma 1, we can assume withoutloss of generality that ϕ is sufficiently large, which we need below. We firstcheck that condition (53) of Theorem 9 is fulfilled. By the Lagrange meanvalue theorem for all ϕ ∈ ( ϕ , ϕ ( ε )), where ϕ ( ε ) is defined in (35), we havethat D := f ( ϕ ) − f (cid:18) ϕ + εf ( ϕ ) + εf ( ϕ ) − ε (cid:19) ≤ | f (cid:48) ( ϕ ) | (cid:18) εf ( ϕ ) + εf ( ϕ ) − ε (cid:19) ≤ M ϕ − α − [log ϕ ] l (cid:18) εmϕ − α [log ϕ ] l + εmϕ − α [log ϕ ] l − ε (cid:19) = ε · M ϕ − [log ϕ ] l (cid:32) m + 1 m − ε · ϕ α [log ϕ ] l (cid:33) . Since ϕ ( ε ) ≤ c · ε − / (1+ α − δ ) , see the proof of Theorem 8, we have ε · ϕ α [log ϕ ] l ≤ ε · ϕ α ≤ ε · ϕ ( ε ) α ≤ c α ε (1 − δ ) / (1+ α − δ ) ≤ m for all ε ∈ (0 , ε ), provided ε is sufficiently small. Therefore,(60) D ≤ ε · M m ϕ [log ϕ ] l < ε, where we assume that ϕ is sufficiently large: ϕ > M /m .The second condition in Theorem 9 is also fulfilled. Indeed, since c · ε − / (1+ α ) ≤ ϕ ( ε ) ≤ c · ε − / (1+ α − δ ) , where δ := sup ε ∈ (0 ,ε ) δ ( ε ) and δ ( ε )being defined as in the proof of Theorem 8, we conclude that mc − α ε α/ (1+ α − δ ) ≤ mc − α ε α/ (1+ α − δ ) (cid:104) log (cid:16) c · ε − / (1+ α − δ ) (cid:17)(cid:105) l ≤ f ( ϕ ( ε )) ≤ mc − α ε α/ (1+ α ) , that is, Cε q ≤ f ( ϕ ( ε )) ≤ Cε − d , where q := α/ (1 + α − δ ). Therefore, byTheorem 9 we have that dim B Γ = dim B (Γ , rad ). Now from this, using (33),Theorems 8 and 9, we obtain21 + α = dim B (Γ , rad ) ≤ dim B Γ ≤ dim B Γ = dim B (Γ , rad ) = 21 + α . This proves (58).(b) To prove (59) it suffices to check that for all ε ∈ (0 , ε ),(61) | Γ( ϕ , ∞ ) ε,rad | − O ( ε ) ≤ | Γ( ϕ , ∞ ) ε | . SCILLATORY INTEGRALS AND FRACTAL DIMENSION 27
Indeed, since Γ( ϕ , ∞ ) ε,rad ⊆ Γ( ϕ , ∞ ) ε ∪ A ( ε ), where A ( ε ) is the area ofthe part of Γ( ϕ , ∞ ) ε corresponding to ϕ < ϕ . This area is clearly of order O ( ε ) since it is contained in the disk B ε ( T ), where T is the point on Γcorresponding to ϕ .From (61) we have M ∗ s (Γ , rad ) ≤ M ∗ s (Γ), for all s ≥
0. From Theorem8 it follows M ∗ d (Γ , rad ) = + ∞ , hence M ∗ d (Γ) = + ∞ . (cid:3) Acknowledgments. This research was supported by: Croatian Science Foun-dation (HRZZ) under the project IP-2014-09-2285, French ANR projectSTAAVF 11-BS01-009, French-Croatian bilateral Cogito project
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