Heisenberg uniqueness pairs for some algebraic curves in the plane
aa r X i v : . [ m a t h . C A ] F e b HEISENBERG UNIQUENESS PAIRS FOR SOMEALGEBRAIC CURVES IN THE PLANE
DEB KUMAR GIRI AND R. K. SRIVASTAVA
Abstract.
A Heisenberg uniqueness pair is a pair (Γ , Λ), where Γ is acurve and Λ is a set in R such that whenever a finite Borel measure µ having support on Γ which is absolutely continuous with respect to thearc length on Γ satisfies ˆ µ | Λ = 0 , then it is identically 0 . In this article,we investigate the Heisenberg uniqueness pairs corresponding to the spiral,hyperbola, circle and certain exponential curves. Further, we work out acharacterization of the Heisenberg uniqueness pairs corresponding to fourparallel lines. In the latter case, we observe a phenomenon of interlacing ofthree trigonometric polynomials. Introduction
The concept of the Heisenberg uniqueness pair has been first introducedin an influential article by Hedenmalm and Montes-Rodrguez (see [7]). Wewould like to mention that Heisenberg uniqueness pair up to a certain extentis similar to an annihilating pair of Borel measurable sets of positive measureas described by Havin and Joricke [6]. Further, the notion of Heisenberguniqueness pair has a sharp contrast to the known results about determiningsets for measures by Sitaram et al. [3, 14], due to the fact that the determiningset Λ for the function ˆ µ has also been considered a thin set.In addition, the question of determining the Heisenberg uniqueness pair fora class of finite measures has also a significant similarity with the celebratedresult due to M. Benedicks (see[1]). That is, support of a function f ∈ L ( R n )and its Fourier transform ˆ f cannot be of finite measure simultaneously. Later,various analogues of the Benedicks theorem have been investigated in differentset ups, including the Heisenberg group and Euclidean motion groups (see[12, 15, 18]).In particular, if Γ is compact, then ˆ µ is a real analytic function havingexponential growth and it can vanish on a very delicate set. Hence in thiscase, finding the Heisenberg uniqueness pairs becomes little easier. However,this question becomes immensely difficult when the measure is supported ona non-compact curve. Eventually, the Heisenberg uniqueness pair is a natural Date : May 12, 2019.2000
Mathematics Subject Classification.
Primary 42A38; Secondary 44A35.
Key words and phrases.
Bessel function, convolution, Fourier transform. invariant to the theme of the well studies uncertainty principle for the Fouriertransform.In the article [7], Hedenmalm and Montes-Rodrguez have shown that thepair (hyperbola, some discrete set) is a Heisenberg uniqueness pair. As a dualproblem, a weak ∗ dense subspace of L ∞ ( R ) has been constructed to solve theKlein-Gordon equation. Further, a complete characterization of the Heisen-berg uniqueness pairs corresponding to any two parallel lines has been givenby Hedenmalm and Montes-Rodrguez (see [7]).Afterward, a considerable amount of work has been done pertaining to theHeisenberg uniqueness pair in the plane as well as in the higher dimensionalEuclidean spaces.Recently, N. Lev [10] and P. Sjolin [16] have independently shown thatcircle and certain system of lines are HUP corresponding to the unit circle S . Further, F. J. Gonzalez Vieli [19] has generalized HUP corresponding to circlein the higher dimension and shown that a sphere whose radius does not lie inthe zero set of the Bessel functions J ( n +2 k − / ; k ∈ Z + , the set of non-negativeintegers, is a HUP corresponding to the unit sphere S n − . Per Sjolin [17] has investigated some of the Heisenberg uniqueness pairscorresponding to the parabola. Subsequently, D. Blasi Babot [2] has given acharacterization of the Heisenberg uniqueness pairs corresponding to a certainsystem of three parallel lines. However, an exact analogue for the finitely manyparallel lines is still open.In a major development, P. Jaming and K. Kellay [8] have given a unifyingproof for some of the Heisenberg uniqueness pairs corresponding to the hyper-bola, polygon, ellipse and graph of the functions ϕ ( t ) = | t | α , whenever α > , via dynamical system approach.Let Γ be a finite disjoint union of smooth curves in R . Let X (Γ) be thespace of all finite complex-valued Borel measure µ in R which is supportedon Γ and absolutely continuous with respect to the arc length measure on Γ.For ( ξ, η ) ∈ R , the Fourier transform of µ is defined byˆ µ ( ξ, η ) = Z Γ e − iπ ( x · ξ + y · η ) dµ ( x, y ) . In the above context, the function ˆ µ becomes a uniformly continuous boundedfunction on R . Thus, we can analyze the pointwise vanishing nature of thefunction ˆ µ. Definition 1.1.
Let Λ be a set in R . The pair (Γ , Λ) is called a Heisenberguniqueness pair for X (Γ) if any µ ∈ X (Γ) satisfying ˆ µ | Λ = 0 , implies µ = 0 . Since the Fourier transform is invariant under translation and rotation, onecan easily deduce the following invariance properties about the Heisenberguniqueness pair.
EISENBERG UNIQUENESS PAIR 3 (i) Let u o , v o ∈ R . Then the pair (Γ , Λ) is a HUP if and only if the pair(Γ + u o , Λ + v o ) is a HUP.(ii) Let T : R → R be an invertible linear transform whose adjoint isdenoted by T ∗ . Then (Γ , Λ) is a HUP if and only if ( T − Γ , T ∗ Λ) is aHUP.Now, we state first known results on the Heisenberg uniqueness pair due toHedenmalm and Montes-Rodrguez [7]. After that, we briefly indicate theprogress on this recent problem.
Theorem 1.2. [7]
Let
Γ = L ∪ L , where L j ; j = 1 , are any two parallelstraight lines and Λ a subset of R such that π (Λ) = R . Then (Γ , Λ) is aHeisenberg uniqueness pair if and only if the set (1.1) e Λ = π a (Λ) ∪ (cid:2) π b (Λ) r π c (Λ) (cid:3) is dense in R . Here we avoid to mention the notations appeared in (1.1) as they are bitinvolved, however, we have written down the same notations as in the article[7]. Though, their main features can be perceived in Section 3.
Theorem 1.3. [7]
Let Γ be the hyperbola x x = 1 and Λ α,β a lattice-crossdefined by Λ α,β = ( α Z × { } ) ∪ ( { } × β Z ) , where α, β are positive reals. Then (Γ , Λ α,β ) is a Heisenberg uniqueness pairif and only if αβ ≤ . For ξ ∈ Λ , define a function e ξ on Γ by e ξ ( x ) = e iπx · ξ . As a dual problem toTheorem 1.3, Hedenmalm and Montes-Rodrguez [7] have proved the followingdensity result which in turn solve the one-dimensional Kein-Gordon equation.
Theorem 1.4. [7]
The pair (Γ , Λ) is a Heisenberg uniqueness pair if and onlyif the set { e ξ : ξ ∈ Λ } is a weak ∗ dense subspace of L ∞ (Γ) . Remark 1.5.
In particular, for Γ to be an algebraic curve, the question ofHeisenberg uniqueness pair can be understood through a partial differentialequation (PDE). That is, if Γ is the zero set of a polynomial P on R , then ˆ µ satisfies the PDE P (cid:18) ∂ iπ , ∂ iπ (cid:19) ˆ µ = 0with initial condition ˆ µ | Λ = 0 . This formulation may help potentially in deter-mining the geometrical structure of the set Z (ˆ µ ) , the zero set of the functionˆ µ. If we consider Λ to be contained in Z (ˆ µ ) , then (Γ , Λ) is not a HUP. Hencethe question of the HUP arises when Λ has located away from Z (ˆ µ ) . In the case when µ is supported on a circle, the function ˆ µ becomes realanalytic and hence it could vanish at most on a very thin set. Thus, there arean enormous number of candidates for Λ such that (Γ , Λ) is a HUP. Some of the
DEB KUMAR GIRI AND R. K. SRIVASTAVA
Heisenberg uniqueness pairs corresponding to circle has been independentlyinvestigated by N. Lev and P. Sjolin. Following are their main results. Formore details, we refer to [10, 16].
Theorem 1.6. [10, 16]
Let
Γ = S be the unit circle. ( i ) Let Λ be a circle of radius r. Then (Γ , Λ) is a HUP if and only if J k ( r ) = 0 for all k ∈ Z + . ( ii ) Let Λ be a straight line. Then (Γ , Λ) is not a HUP. ( iii ) Let
Λ = L ∪ L , where L j ; j = 1 , are two straight lines. If L and L are parallel, then (Γ , Λ) is a HUP. ( iv ) Let L j ; j = 1 , , . . . , N be the N different straight lines which intersect atone point and angle between any of two lines out of these N lines is of the form πα. Let Λ N = N S j =1 L j . Then (Γ , Λ N ) is not a HUP if and only if α is rational. In contrast to the case of finitely many straight lines, P. Sjolin [16] has shownthat if Λ = ∞ S k =1 L k , where { L k } is a sequence of straight lines which intersectat one point. Then ( S , Λ) is a HUP.
Remark 1.7.
Since we know that any homogeneous harmonic polynomialon R can be expressed as Ar j sin( jθ + δ ) for some j ∈ N and δ ∈ [0 , π )(see [5]), up to some rotation and translation, we can think of Λ N = N S k =1 L k , appeared in Theorem 1.6 ( iv ) , as the zero set of some homogeneous harmonicpolynomial. If ( S , Λ) is a Heisenberg uniqueness pair, then the set Λ must beaway from the zero set of any homogeneous harmonic polynomial. However,the converse is not true. Since ( S , Λ) is not a HUP if Λ is a circle whoseradius lie in the zero set of some Bessel function. Thus, it is an interestingquestion to examine the exceptional sets for the Heisenberg uniqueness pairscorresponding to circle.Subsequently, some of the Heisenberg uniqueness pairs corresponding to theparabola have been obtained by P. Sjolin [17]. Let | E | denotes the Lebesguemeasure of the set E ⊂ R . Theorem 1.8. [17]
Let Γ denote the parabola y = x . ( i ) Let
Λ = L be a straight line. Then (Γ , Λ) is a HUP if and only if L isparallel to the X-axis. ( ii ) Let
Λ = L ∪ L , where L j ; j = 1 , are two different straight lines. Then (Γ , Λ) is a HUP. ( iii ) Let L j ; j = 1 , be two different straight lines which are not parallel tothe X -axis. Let E j ⊂ L j and | E j | > j = 1 , . If Λ = E ∪ E , then (Γ , Λ) isa HUP. EISENBERG UNIQUENESS PAIR 5
The question of Heisenberg uniqueness pair in the higher dimension hasbeen first taken up by F. J. Gonzalez Vieli [19, 20].
Theorem 1.9. [19]
Let
Γ = S n − be the unit sphere in R n and Λ a sphere ofradius r. Then (Γ , Λ) is a HUP if and only if J ( n +2 k − / ( r ) = 0 for all k ∈ Z + . Theorem 1.10. [20]
Let Γ be the paraboloid x n = x + x + · · · + x n − in R n and Λ an affine hyperplane in R n of dimension n − . Then (Γ , Λ) is a HUPif and only if Λ is parallel to the hyperplane x n = 0 . Let Γ denote a system of three parallel lines in the plane that can be ex-pressed as Γ = R × { α, β, γ } , where α < β < γ and ( γ − α ) / ( β − α ) ∈ N . By the invariance properties of HUP, one can assume that Γ = R × { , , p } , for some p ∈ N with p ≥ . The following characterization for the Heisenberguniqueness pairs corresponding to the above mentioned three parallel lines hasbeen given by D. B. Babot [2].
Theorem 1.11. [2]
Let
Γ = R × { , , p } , for some p ∈ N with p ≥ and Λ ⊂ R a closed set which is -periodic with respect to the second variable.Then (Γ , Λ) is a HUP if and only if the set (1.2) e Λ = Π (Λ) ∪ (cid:2) Π (Λ) r Π ∗ (Λ) (cid:3) ∪ (cid:2) Π (Λ) r Π ∗ (Λ) (cid:3) is dense in R . For the notations appeared in Equation (1.2), we would like to refer thearticle [2], as those notations are quite involved. However, the nature of theiroccurrence can be understood in the beginning of Section 3 when we formulatethe four lines problem.Further, Jaming and Kellay [8] have given a unifying proof for some of theHeisenberg uniqueness pairs corresponding to certain algebraic curves.
Theorem 1.12. [8]
Let Γ be any of the following curves:(i) the graph of ψ ( t ) = | t | α , t ∈ R , α > (ii) a hyperbola;(iii) a polygon;(iv) an ellipse.Then there exists a set E ⊂ ( − π/ , π/ × ( − π/ , π/ of positive Lebesguemeasure such that for ( θ , θ ) ∈ E, the pair (Γ , L θ ∪ L θ ) is a HUP. A review of the Heisenberg uniqueness pairs for the spiral,hyperbola, circle and exponential curves
In this section, we will work out some of the Heisenberg uniqueness pairscorresponding to the spiral, hyperbola, circle and certain exponential curvesby using the basic tools of the Fourier analysis. Though, a complete charac-terization for the Heisenberg uniqueness pairs corresponding to either of theabove curves is still open.
DEB KUMAR GIRI AND R. K. SRIVASTAVA
First, we prove that the spiral is a Heisenberg uniqueness pair for the anti-spiral.
Theorem 2.1.
Suppose
Γ = { ( e − t cos t, e − t sin t ) : t ≥ } is a spiral and let Λ = { ( e s cos s, e s sin s ) : s ≤ } . Then (Γ , Λ) is a Heisenberg uniqueness pair. In order to prove Theorem 2.1, we need the following results from [3, 4].
Theorem 2.2. [4]
Let h be a bounded measurable function and g ∈ L ( R n ) . If h ∗ g vanishes identically, then ˆ h vanishes on the support of ˆ g. Let R n + = { ( x , . . . , x n ) ∈ R n : x j ≥ j = 1 , . . . , n } . The following resulthad appeared in the article [3] by Bagchi and Sitaram, p. 421, as a part of theproof of Proposition 2 . . Proposition 2.3. [3]
Let h be a non-zero bounded Borel measurable functionwhich is supported on R n + . Then supp ˆ h = R n . Proof of Theorem 2.1.
Since µ is absolutely continuous with respect to the arclength measure on Γ , by Radon-Nikodym theorem there exists f ∈ L [0 , ∞ )such that dµ = √ f ( t ) e − t dt. Let g ( t ) = √ f ( t ) e − t . Then by the finiteness of µ, it follows that g ∈ L [0 , ∞ ) . By hypothesis, ˆ µ | Λ = 0 implies(2.1) ˆ µ ( ξ, η ) = Z ∞ e − iπe − t ( ξ cos t + η sin t ) dµ ( t ) = Z ∞ e − iπe ( s − t ) cos( t − s ) g ( t ) dt = 0for all ( ξ, η ) ∈ Λ . Let H ( t ) = e − iπe t cos t χ [0 , ∞ ) ( t ) and G ( t ) = g ( t ) χ (0 , ∞ ) ( t ) . Thenfrom (2.1), we get ˆ µ ( ξ, η ) = ( H ∗ G )( s ) = 0 ∀ s ∈ R . In view of Theorem2.2, we infer that supp ˆ H ⊂ Z ( ˆ G ) , where Z ( ˆ G ) denotes the zero set of ˆ G. As H is a non-zero bounded Borel measurable function supported in [0 , ∞ ) , byProposition 2.3 it follows that supp ˆ H = R and hence ˆ G = 0 . Thus, µ = 0 . Next, we work out some of the Heisenberg uniqueness pairs correspondingto certain exponential curves in the plane. Though, the result is true for alarge class of exponential curves, for the sake of simplicity we prove only for aparticular one.
Theorem 2.4.
Let α : R → R + be the function defined by α ( t ) = e t and let Γ = { ( t, α ( t )) : t ∈ R } . ( i ) If Λ is a straight line parallel to the X -axis. Then (Γ , Λ) is a HUP. ( ii ) Let
Λ = L ∪ L , where L j ; j = 1 , are any two straight lines parallel tothe Y -axis. Then (Γ , Λ) is a HUP. In order to prove Theorem 2.4, we need the following two important resultsabout the uniqueness of Fourier transform. First, we state a result which canbe found in Havin and Joricke [6], p. 36.
Lemma 2.5. [6] If ϕ ∈ L ( R ) is supported in [0 , ∞ ) and R R log | ˆ ϕ | dx x = −∞ , then ϕ = 0 . EISENBERG UNIQUENESS PAIR 7
As a consequence of Lemma 2.5 we prove the following result.
Lemma 2.6.
Let g ∈ L ( R ) and α : R → R + be defined by α ( t ) = e t . Suppose E ⊂ R and | E | > . Then (2.2) Z R e − iπyα ( t ) g ( t ) dt = 0 for all y ∈ E if and only if g is an odd function.Proof. The left hand side of Equation (2.2) can be expressed as I = Z −∞ e − iπyα ( t ) g ( t ) dt + Z ∞ e − iπyα ( t ) g ( t ) dt = Z ∞ e − iπyα ( t ) ( g ( t ) + g ( − t )) dt = Z ∞ e − iπyα ( t ) F ( t ) dt, where F ( t ) = g ( t ) + g ( − t ) for all t ≥ . Clearly F ∈ L (0 , ∞ ) and hence bythe change of variables u = α ( t ) , we have(2.3) I = Z ∞ e − iπxu F ( p log u ) du u √ log u . Let ϕ ( u ) = F ( √ log u ) / u √ log u χ (1 , ∞ ) ( u ) . Then ϕ ∈ L ( R ) and from (2.3) wehave I = ˆ ϕ ( y ) = 0 for all y ∈ E. Since ˆ ϕ vanishes on the set E of positiveLebesgue measure, by Lemma 2.5 it follows that ϕ = 0 . That is, F = 0 andhence g is an odd function.Conversely, if g is an odd function, then (2.2) trivially holds. (cid:3) Proof of Theorem 2.4. ( i ) Since µ is supported on Γ = { ( t, e t ) : t ∈ R } , there exists f ∈ L ( R ) such that dµ = f ( t ) √ t e t dt . Let g ( t ) = f ( t ) √ t e t . Then by the finiteness of µ, it follows that g ∈ L ( R ) andˆ µ ( x, y ) = Z R e − iπ (cid:16) xt + ye t (cid:17) g ( t ) dt. In view of the invariance property ( i ) , we can assume that Λ is the X -axis.Hence ˆ µ | Λ = 0 implies that ˆ g ( x ) = 0 for all x ∈ R . Thus, we conclude that µ = 0 . ( ii ) By invariance property ( i ) , we can assume L is the Y -axis and L the line x = x o , where x o = 0 . Since ˆ µ vanishes on L , by Lemma 2.6 it follows that g is odd. Also, ˆ µ vanishes on the line L , implies that Z R e − iπ ( x o t + ye t ) g ( t ) dt = 0for all y ∈ R . In view of Lemma 2.6, it follows that e − iπx o t g ( t ) is an oddfunction. Hence e − iπx o t g ( t ) = − e iπx o t g ( − t ) . Since g is odd, it implies that DEB KUMAR GIRI AND R. K. SRIVASTAVA ( e iπx o t − g ( t ) = 0 . As the identity e iπx o t = 1 holds only for the countablymany t, we conclude that g = 0 . Thus, µ = 0 . Remark 2.7.
Let α : R → R + be an even smooth function having finitelymany local extrema and Γ = { ( t, α ( t )) : t ∈ R } . Then the conclusions of The-orem 2.4 would also hold.Next, we work out some of the Heisenberg uniqueness pairs correspondingto the circle. We show that (circle, spiral) is a HUP.Let Γ = S denote the unit circle in R . If for f ∈ L (Γ) , we write f ( θ )instead of f ( e iθ ) , then f is a 2 π periodic function and f ∈ L [0 , π ). Let µ be a finite complex-valued Borel measure in R which is supported on Γ andabsolutely continuous with respect to the arc length measure on Γ. Then thereexists f ∈ L ( S ) such that dµ = f ( θ ) dθ . Now, we prove the following result. Theorem 2.8.
Let
Γ = S and Λ = { ( e t cos t, e t sin t ) : t ≤ } be the spiral.Then (Γ , Λ) is a Heisenberg uniqueness pair.Proof. Since µ is supported on the unit circle Γ , we can write the Fouriertransform of µ by ˆ µ ( x, y ) = Z π − π e − iπ ( x cos θ + y sin θ ) f ( θ ) dθ. Hence ˆ µ can be extended holomorphically to C . Thus, the function F definedby F ( z , z ) = Z π − π e − iπ ( z cos θ + z sin θ ) f ( θ ) dθ, is holomorphic on C . In particular, ˆ µ = F | R is a real analytic function. Sinceˆ µ vanishes on the spiral Λ , for any line L which passes through the origin,ˆ µ | Λ ∩ L = 0 . As (0 ,
0) is a limit point of the set Λ ∩ L, it follows that ˆ µ | L = 0 . Since L is arbitrary, we infer that ˆ µ ( x, y ) = 0 for all ( x, y ) ∈ R . Let S r = { ( r cos t, r sin t ) : 0 ≤ t < π } , where J k ( r ) = 0 for all k ∈ Z . Thenˆ µ ( r cos t, r sin t ) = 0 implies h ∗ f ( t ) = 0 , where h ( t ) = e − iπr cos t . As we knowthat the Fourier coefficients of h satisfying ˆ h ( k ) = i k ( − k J k ( r ) , it follows thatˆ f ( k ) J k ( r ) = 0 for all k ∈ Z . Since J k ( r ) = 0 for all k ∈ Z , ˆ f ( k ) = 0 for all k ∈ Z and hence f = 0 . (cid:3) Remark 2.9.
A set which is determining set for any real analytic functionis called
N A - set. For instance, the spiral is an
N A - set in the plane (see[13]). If µ is a finite Borel measure supported on a closed and bounded curveΓ , then ˆ µ is real analytic. Thus, (Γ , NA - set) is a Heisenberg uniqueness pair.However, the converse is not true.Hence, in view of Remarks 1.7 and 2.9 we expect that the exceptional setsfor the Heisenberg uniqueness pairs corresponding to the unit circle Γ = S areeventually contained in the zero sets of all homogeneous harmonic polynomialsunion with the countably many circles whose radii are lying in the zero set of EISENBERG UNIQUENESS PAIR 9 the certain class of Bessel functions. On the basis of these credible observa-tions, we are trying to find out a complete characterization of the Heisenberguniqueness pairs corresponding to circle which may be presented somewhereelse.Next, we work out some of the Heisenberg uniqueness pairs correspondingto the hyperbola. Though in this case, Hedenmalm and Montes-Rodrguez[7] have found that some discrete set Λ α,β is enough for (Γ , Λ α,β ) to be aHeisenberg uniqueness pair. However, our approach is to consider those setsΛ which are essentially a union of continuous curves and located somewhereelse than the set Λ α,β . Theorem 2.10.
Let
Γ = { (cosh t, sinh t ) : t ≥ } be a branch of the hyperbolaand Λ = { (cosh s, − sinh s ) : s ∈ R } . Then (Γ , Λ) is a HUP.Proof. Since µ is supported on Γ , there exists f ∈ L [0 , ∞ ) such that dµ = f ( t ) √ cosh 2 t dt . If we write g ( t ) = f ( t ) √ cosh 2 t, then g ∈ L [0 , ∞ ) andˆ µ ( x, y ) = Z ∞ e − iπ ( x cosh t + y sinh t ) g ( t ) dt. By hypothesis, ˆ µ | Λ = 0 implies(2.4) ˆ µ ( x, y ) = Z ∞ e − iπ cosh( t − s ) g ( t ) dt = 0for all ( x, y ) ∈ Λ . Let H ( t ) = e − iπ cosh t χ [0 , ∞ ) ( t ) and G ( t ) = g ( t ) χ (0 , ∞ ) ( t ) . Thenfrom (2.4) we get ˆ µ ( x, y ) = ( H ∗ G )( s ) = 0 for all s ∈ R . In view of Theorem2.2, it follows that supp ˆ H ⊂ Z ( ˆ G ) . Hence by Proposition 2.3, supp ˆ H = R . Thus, we conclude that G = 0 . (cid:3) Theorem 2.11.
Let
Γ = { (cosh t, sinh t ) : t ∈ R } and Λ = L ∪ L , where L j ; j = 1 , are any two lines parallel to the X -axis. Then (Γ , Λ) is a HUP. We need the following result in order to prove Theorem 2.11.
Lemma 2.12.
Let g ∈ L ( R ) and E ⊂ R such that | E | > . Then (2.5) Z R e − iπx cosh t g ( t ) dt = 0 for all x ∈ E if and only if g is an odd function.Proof. The left-hand side of Equation (2.5) can be expressed as I = Z −∞ e − iπx cosh t g ( t ) dt + Z ∞ e − iπx cosh t g ( t ) dt = Z ∞ e − iπx cosh t ( g ( t ) + g ( − t )) dt = Z ∞ e − iπx cosh t F ( t ) dt, where F ( t ) = g ( t ) + g ( − t ) for all t ≥ . Clearly F ∈ L (0 , ∞ ) . By change ofvariables u = cosh t, we get(2.6) I = Z ∞ e − iπxu F (cosh − u ) du √ u − . If we substitute ϕ ( u ) = F (cosh − u ) / √ u − χ (1 , ∞ ) , then ϕ ∈ L ( R ) and I = ˆ ϕ ( x ) = 0 for all x ∈ E. Hence by Lemma 2.5, it follows that ϕ = 0 . Thus,we infer that g is an odd function.Conversely, suppose g is an odd function, then (2.5) trivially holds. (cid:3) Proof of Theorem 2.11.
By invariance property ( i ) , we can assume that L is the X -axis and L the line y = y o , where y o = 0 . Since µ is supported onthe hyperbola Γ , there exists f ∈ L ( R ) such that dµ = f ( t ) √ cosh 2 t dt . Let g ( t ) = f ( t ) √ cosh 2 t, then g ∈ L ( R ) . Hence in view of Lemma 2.12, ˆ µ vanisheson L implies that g is an odd function. Further, ˆ µ | L = 0 implies that Z R e − iπ ( x cosh t + y o sinh t ) g ( t ) dt = 0for all x ∈ R . Then by Lemma 2.12 the function e − iπy o sinh t g ( t ) will be an oddfunction. Hence e − iπy o sinh t g ( t ) = − e iπy o sinh t g ( − t ) . As g is an odd function, itfollows that (cid:0) e iπy o sinh t − (cid:1) g ( t ) = 0 . Using the fact that e iπy o sinh t = 1 holdsonly for the countably many values of t, we conclude that g = 0 . Theorem 2.13.
Let
Γ = { (cosh t, sinh t ) : t ∈ R } and Λ = L ∪ L , where L j ; j = 1 , are any two straight lines which intersect at an angle α ∈ (0 , π ) . Then (Γ , Λ) is a HUP.Proof. Without loss of generality, we can assume that L is the X -axis and L = { ( s cosh t o , − s sinh t o ) : s ∈ R } , where tan α = − tanh t o . Since µ issupported on the hyperbola Γ , as similar to Theorem 2.11 there exists g ∈ L ( R ) such that dµ = g ( t ) dt. Suppose ˆ µ = 0 on Λ , then we have Z R e − iπ ( x cosh t + y sinh t ) g ( t ) dt = 0for all ( x, y ) ∈ L . This in turn implies Z R e − iπs cosh t g ( t + t o ) dt = 0for all s ∈ R . In view of Lemma 2.12, it follows that g ( t o + · ) must be anodd function. Since ˆ µ is also vanishing on the X -axis, g will be odd. Hence g (2 t o ± t ) = g ( t ) for all t ∈ R . That is, g is a periodic function contained in L ( R ) . Thus, we conclude that g = 0 . (cid:3) Remark 2.14. ( a ) . Let Γ be the hyperbola and Λ a straight line parallel tothe X -axis. Then (Γ , Λ) is not a HUP. Consider g = √ cosh 2 t sin t χ ( − π,π ) and dµ = g ( t ) dt. Then ˆ µ vanishes on Λ . EISENBERG UNIQUENESS PAIR 11 ( b ) . We would like to mention that Theorem 2.13 is contained well in the case( ii ) of Theorem 1.12 due to Jaming and Kellay [8]. However, our approach forproof of Theorem 2.13 is quite different.3. Heisenberg uniqueness pairs corresponding to the fourparallel lines
A characterization of the Heisenberg uniqueness pairs corresponding to anytwo parallel straight lines have been done by Hedenmalm et al. [7]. Further, D.B. Babot [2] has worked out an analogous result for a certain system of threeparallel lines. In this section, we prove a characterization of the Heisenberguniqueness pairs corresponding to a certain system of four parallel lines. Inthe above case, we observe the phenomenon of interlacing of three totallydisconnected sets.Let Γ o denote a system of four parallel lines that can be expressed as Γ o = R × { α, β, γ, δ } , where α < β < γ < δ, p = ( δ − α ) / ( β − α ) ∈ N r { , } and( γ − α ) / ( β − α ) = 2 . If (Γ o , Λ o ) is a HUP, then by using invariance property( i ) , (Γ o , Λ o ) can be reduced to (Γ o − (0 , α ) , Λ o ) . Since scaling can be thoughtas a diagonal matrix, by using invariance property ( ii ) , (Γ o − (0 , α ) , Λ o ) canbe reduced to ( T − (Γ o − (0 , α )) , T ∗ Λ o ) , where T = diag { ( β − α ) , ( β − α ) } . LetΛ = T ∗ Λ o and Γ = T − (Γ o − (0 , α )) . Then Γ = R × { , , , p } , where p ∈ N with p ≥ . Thus, (Γ o , Λ o ) is a HUP if and only if (Γ , Λ) is a HUP.Before we state our main result of this section, we need to set up somenecessary notations and the subsequent auxiliary results.Let µ be a finite Borel measure which is supported on Γ and absolutelycontinuous with respect to the arc length measure on Γ . Then there existfunctions f k ∈ L ( R ); k = 0 , , , dµ = f ( x ) dxdδ ( y ) + f ( x ) dxdδ ( y ) + f ( x ) dxdδ ( y ) + f ( x ) dxdδ p ( y ) , where δ t denotes the point mass measure at t. By taking the Fourier transformof both sides of (3.1) we get(3.2) ˆ µ ( ξ, η ) = ˆ f ( ξ ) + e πiη ˆ f ( ξ ) + e πiη ˆ f ( ξ ) + e pπiη ˆ f ( ξ ) . Notice that for each fixed ( ξ, η ) ∈ Λ , the right-hand side of Equation (3 .
2) isa trigonometric polynomial of degree p that could have preferably some missingterms. Therefore, it is an interesting question to find out the smallest set Λthat determines the above trigonometric polynomial. We observe that the sizeof Λ depends on the choice of a number of lines as well as irregular separationamong themselves. That is, a larger number of lines or value of p would forcesmaller size of Λ . Eventually, the problem would become immensely difficultfor a large value of p. Observe that ˆ µ is a 2-periodic function in the second variable. Hence, forany set Λ ⊂ R , it is enough to consider the set £ (Λ) = { ( ξ, η ) : ( ξ, η + 2 k ) ∈ Λ , for some k ∈ Z } for the purpose of HUP. Also, it is easy to verify that (Γ , Λ) is a HUP if andonly if (Γ , £ (Λ)) is a HUP, where £ (Λ) denotes the closure of £ (Λ) in R . Inview of the above facts, it is enough to work with the closed set Λ ⊂ R whichis 2-periodic with respect to the second variable.Now, it is evident from the Riemann-Lebesgue lemma that the exponentialfunctions, which appeared in (3.2), cannot be expressed as the Fourier trans-form of functions in L ( R ) . However, they can locally agree with the Fouriertransform of functions in L ( R ) . Hence, in view of the condition ˆ µ | Λ = 0 , wecan classify these related exponential functions.Given a set E ⊂ R and a point ξ ∈ E, let I ξ denote an interval containing ξ. We define three functions spaces in the following way. (A). L E,ξloc = { ψ : E → C such that ψ ( ξ ) = 0 and there is an interval I ξ and afunction ϕ ∈ L ( R ) which satisfies ψ = ˆ ϕ on I ξ ∩ E } . (B). P , [ L E,ξloc ] = { ψ : E → C such that there is an interval I ξ and ϕ j ∈ L ( R ); j = 0 , ψ + ˆ ϕ ψ + ˆ ϕ = 0 on I ξ ∩ E } . Now, for p ∈ N with p ≥ , we define the third functions space as follows. (C). P ,p [ L E,ξloc ] = { ψ : E → C such that there is an interval I ξ and functions ϕ j ∈ L ( R ); j = 0 , , ψ p + ˆ ϕ ψ + ˆ ϕ ψ + ˆ ϕ = 0 on I ξ ∩ E } . We will frequently use the following Wiener’s lemma that plays a key rolein the rest part of the arguments for proofs.
Lemma 3.1. [9]
Let ψ ∈ L E,ξloc and ψ ( ξ ) = 0 . Then /ψ ∈ L E,ξloc . For more details, see [9], p.57.In view of Lemma 3.1, we derive the following relation among the sets whichare described by (A) , (B) and (C) . We would like to mention that the integralchoice of p in Lemma 3.2 has been considered for a convenience. Lemma 3.2.
For p ≥ , the following inclusions hold. (3.3) L E,ξloc ⊂ P , [ L E,ξloc ] ⊂ P ,p [ L E,ξloc ] . Proof. ( a ) If ψ ∈ L E,ξloc , then by the Wiener’s lemma 1 /ψ ∈ L E,ξloc . By definition,there exist intervals I , I containing ξ and functions f, g ∈ L ( R ) such that ψ = ˆ f on I ∩ E and ψ = ˆ g on I ∩ E. Hence we can extract an interval I ⊂ I ∩ I containing ξ such that ψ = ˆ f ˆ g on I ∩ E. As ˆ g ( ξ ) = 0 , thereexists an interval I containing ξ and a function h ∈ L ( R ) such that g = ˆ h on I ∩ E. Further, we can extract an interval I ⊂ I ∩ I containing ξ such that(3.4) ψ = ˆ f ˆ h = [ f ∗ h = ˆ ϕ on I ∩ E, where ϕ = f ∗ h ∈ L ( R ) . This implies ψ ∈ L E,ξloc . Hence by theinduction argument, it can be shown that ψ p ∈ L E,ξloc , whenever p ∈ N . Now,
EISENBERG UNIQUENESS PAIR 13 consider a function f ∈ L ( R ) such that I ⊂ supp ˆ f . Since ψ = ˆ f on I ∩ E, it follows that(3.5) ˆ f ψ = ˆ f ˆ f = \ f ∗ f on I ∩ E. Hence from (3.4) and (3.5) we conclude that(3.6) ψ + ˆ ϕ ψ + ˆ ϕ = 0on I ξ ∩ E, where I ξ ⊂ I ∩ I , ϕ = − ( f ∗ f + ϕ ) and ϕ = f . Thus, ψ ∈ P , [ L E,ξloc ] . By applying induction, we can show that ψ p + ˆ ϕ ψ + ˆ ϕ = 0 , whenever p ∈ N . ( b ) If ψ ∈ P , [ L E,ξloc ] , then there exists an interval I ξ containing ξ and functions f, g ∈ L ( R ) such that(3.7) ψ + ˆ f ψ + ˆ g = 0on I ξ ∩ E. Now, consider a function f ∈ L ( R ) such that I ξ ⊂ supp ˆ f . Aftermultiplying (3.7) by ψ and ˆ f separately and adding the resultant equations,we can write ψ + (cid:16) ˆ f + ˆ f (cid:17) ψ + (cid:16) ˆ f ˆ f + ˆ g (cid:17) ψ + ˆ f ˆ g = 0 . Hence for the appropriate choice of ϕ j ; j = 0 , , , we have(3.8) ψ + ˆ ϕ ψ + ˆ ϕ ψ + ˆ ϕ = 0on I ξ ∩ E. Further by induction, it follows that ψ p + ˆ ϕ ψ + ˆ ϕ ψ + ˆ ϕ = 0 on I ξ ∩ E, whenever p ∈ N . Thus, ψ ∈ P ,p [ L E,ξloc ] . (cid:3) Let Π(Λ) be the projection of Λ on R × { } . For ξ ∈ Π(Λ) , we denote thecorresponding image on the η - axis by Σ ξ = { η ∈ [0 ,
2) : ( ξ, η ) ∈ Λ } . Now, we require analyzing the set Π(Λ) to know its basic geometrical struc-ture in accordance with the Heisenberg uniqueness pair. Since it is expectedthat the set Σ ξ may consist one or more image points depending upon theorder of its winding, the set Π(Λ) can be decomposed into the following fourdisjoint sets. For the sake of convenience, we denote F o = { , , , } . ( P ) . Π (Λ) = { ξ ∈ Π(Λ) : there is a unique η ∈ Σ ξ } . ( P ) . Π (Λ) = { ξ ∈ Π(Λ) : there are only two distinct η j ∈ Σ ξ ; j = 0 , } . In order to describe the rest of the two partitioning sets, we will use thenotion of symmetric polynomial. For each k ∈ Z + , the complete homogeneoussymmetric polynomial H k of degree k is the sum of all monomials of degree k. That is, H k ( x , . . . , x n ) = X l + ··· + l n = k ; l i ≥ x l . . . x l n n . For more details, we refer to [11].
Consider four distinct image points η j ∈ [0 ,
2) and denote a j = e πiη j ; j ∈ F . For p ≥ , we define the remaining two sets as follows:( P ) . Π (Λ) = { ξ ∈ Π(Λ) : there are at least three distinct η j ∈ Σ ξ for j =0 , , η ∈ Σ ξ , then H p − ( a , a , a ) = H p − ( a , a , a ) } . ( P ) . Π (Λ) = { ξ ∈ Π(Λ) : there are at least four distinct η j ∈ Σ ξ ; j ∈ F o which satisfy H p − ( a , a , a ) = H p − ( a , a , a ) } . In this way, we get the desired decomposition as Π(Λ) = S j =1 Π j (Λ) . Now, for three distinct image points η j ∈ [0 , j = 0 , , , denote a = e πiη ,b = e πiη and c = e πiη . Consider the system of equations A ξ X = B pξ , where(3.9) A ξ = a a b b c c ,X ξ = ( τ , τ , τ ) and B pξ = − ( a p , b p , c p ) . Since det A ξ = ( a − b )( b − c )( c − a ) = 0 ,A ξ X = B pξ has a unique solution. A simple calculation gives τ o = − abcH p − ( a, b, c ) , (3.10) τ = H p − ( a, b, c ) − (cid:0) a p − + b p − + c p − (cid:1) + X l + m + n = p − l,m,n ≥ a l b m c n ,τ = − H p − ( a, b, c ) . Since measure in the question is supported on a certain system of four paral-lel lines and the exponential functions which have appeared in (3.2) can locallyagree with the Fourier transform of some functions in L ( R ) , the following setssitting in Π(Λ) seems to be dispensable in the process of getting the Heisenberguniqueness pairs.( P ∗ ) . As each ξ ∈ Π (Λ) has a unique image in Σ ξ , we can define a function χ on Π (Λ) by χ ( ξ ) = e πiη , where η = η ( ξ ) ∈ Σ ξ . Now, the first dispensableset can be defined byΠ ∗ (Λ) = n ξ ∈ Π (Λ) : χ ∈ P ,p [ L Π (Λ) ,ξloc ] o . Next, for ξ ∈ Π (Λ) , let χ j ( ξ ) = e πiη j , where η j = η j ( ξ ) ∈ Σ ξ ; j = 0 , . ( P ′ ∗ ) . Since each ξ ∈ Π (Λ) has two distinct image points in Σ ξ , we definetwo functions δ j on Π (Λ); j = 0 , X ξ = ( δ ( ξ ) , δ ( ξ )) is the solutionof A ξ X ξ = B ξ , where A ξ = (cid:18) χ ( ξ )1 χ ( ξ ) (cid:19) EISENBERG UNIQUENESS PAIR 15 and B ξ = − (cid:0) χ ( ξ ) , χ ( ξ ) (cid:1) . In this way, an auxiliary dispensable set can bedefined by Π ∗ (Λ) = n ξ ∈ Π (Λ) : δ j ∈ L Π (Λ) ,ξloc ; j = 0 , o . ( P ′′ ∗ ) . Further, we define three functions ρ j on Π (Λ); j = 0 , , X ξ = ( ρ ( ξ ) , ρ ( ξ ) , ρ ( ξ )) becomes a solution of A pξ X ξ = B pξ , where A pξ = (cid:18) χ ( ξ ) χ ( ξ ) χ ( ξ ) χ ( ξ ) (cid:19) and B pξ = − ( χ ( ξ ) p , χ ( ξ ) p ) . Hence the second dispensable set can be definedby Π p ∗ (Λ) = n ξ ∈ Π (Λ) : ρ j ∈ L Π (Λ) ,ξloc ; j = 0 , , o . For ξ ∈ Π (Λ) , let χ j ( ξ ) = e πiη j , where η j = η j ( ξ ) ∈ Σ ξ ; j = 0 , , . ( P ′ ∗ ) . For each ξ ∈ Π (Λ) has three distinct image points in Σ ξ , we definethree functions e j on Π (Λ); j = 0 , , X ξ = ( e ( ξ ) , e ( ξ ) , e ( ξ )) isthe solution of A ξ X ξ = B ξ , where A ξ is the matrix given by Equation (3.9)and B ξ = − (cid:0) χ ( ξ ) , χ ( ξ ) , χ ( ξ ) (cid:1) . Hence another auxiliary dispensable setcan be defined byΠ ∗ (Λ) = n ξ ∈ Π (Λ) : e j ∈ L Π (Λ) ,ξloc ; j = 0 , , o . ( P ′′ ∗ ) . Once again we define three functions τ j on Π (Λ); j = 0 , , X ξ = ( τ ( ξ ) , τ ( ξ ) , τ ( ξ )) is the solution of A ξ X ξ = B pξ , where B pξ = − ( χ ( ξ ) p , χ ( ξ ) p , χ ( ξ ) p ) . Hence the third dispensable set can be defined byΠ p ∗ (Λ) = n ξ ∈ Π (Λ) : τ j ∈ L Π (Λ) ,ξloc ; j = 0 , , o . Now, we prove the following two lemmas that speak about a sharp contrastin the pattern of dispensable sets as compared to dispensable sets which ap-peared in two lines and three lines results. That is, a larger value of p willincrease the size of dispensable sets in case of four lines problem. Further,we observe that dispensable sets are eventually those sets contained in Π(Λ)where we could not solve Equation (3.2). For more details, we refer to [2, 7]. Lemma 3.3.
For p ≥ , the following inclusion holds. Π ∗ (Λ) ⊂ Π p ∗ (Λ) . Proof. If ξ o ∈ Π ∗ (Λ) , then δ j ∈ L Π (Λ) ,ξ o loc . Hence there exists an interval I ξ o containing ξ o and ϕ j ∈ L ( R ) such that δ j = ˆ ϕ j ; j = 0 , ϕ + ˆ ϕ χ j + χ j = 0 on I ξ o ∩ Π (Λ) , whenever j = 0 , . Now, by the similar iteration as in the proofof Lemma 3.2( b ) , we infer that there exist a common set of ψ j ∈ L ( R ); j =0 , , ψ + ˆ ψ χ j + ˆ ψ χ j + χ j p = 0on I ξ o ∩ Π (Λ) , whenever j = 0 , . If we denote ˆ ψ j = ρ j , then it is easy to seethat ξ o ∈ Π p ∗ (Λ) . (cid:3) Lemma 3.4.
For p ≥ , the following inclusion holds. Π ∗ (Λ) ⊆ Π p ∗ (Λ) . Moreover, equality holds for p = 3 . Proof. If ξ o ∈ Π ∗ (Λ) , then e j ∈ L Π (Λ) ,ξ o loc . Hence there exists an interval I ξ o containing ξ o and ϕ j ∈ L ( R ) such that e j = ˆ ϕ j ; j = 0 , , ϕ + ˆ ϕ χ j + ˆ ϕ χ j + χ j = 0on I ξ o ∩ Π (Λ) , whenever j = 0 , , . By the similar iteration as in the proof ofLemma 3.2( b ) , it follows that there exist ψ j ∈ L ( R ); j = 0 , , ψ + ˆ ψ χ j + ˆ ψ χ j + χ j p = 0on I ξ o ∩ Π (Λ) , whenever j = 0 , , . If we denote ˆ ψ j = τ j , then it is easy toverify that ξ o ∈ Π p ∗ (Λ) . (cid:3) On the basis of structural properties of the dispensable sets, we observe thatthese sets are essentially minimizing the size of projection Π(Λ) . Now, we canstate our main result of this section about the Heisenberg uniqueness pairscorresponding to the above described system of four parallel straight lines.
Theorem 3.5.
Let
Γ = R × { , , , p } , where p ∈ N and p ≥ . Let Λ ⊂ R be a closed set which is -periodic with respect to the second variable. Suppose Π(Λ) is dense in R . If (Γ , Λ) is a Heisenberg uniqueness pair, then the set e Π(Λ) = Π (Λ) [ j =0 h Π (3 − j ) (Λ) r Π (3 − j ) ∗ (Λ) i is dense in R . Conversely, if the set f Π p (Λ) = Π (Λ) ∪ (cid:2) Π (Λ) r Π p ∗ (Λ) (cid:3) ∪ (cid:2) Π (Λ) r Π p ∗ (Λ) (cid:3) ∪ (cid:2) Π (Λ) r Π ∗ (Λ) (cid:3) is dense in R , then (Γ , Λ) is a Heisenberg uniqueness pair. Remark 3.6.
In view of Lemma 3.3, we infer that Π ∗ (Λ) is a proper subset ofΠ p ∗ (Λ) for any p ≥ . However, for p = 3 , Lemma 3.4 yields Π ∗ (Λ) = Π p ∗ (Λ) . Hence for any p ≥ , the set f Π p (Λ) is properly contained in e Π(Λ) . Thus, ananalogous result for four lines problem as compared to three lines result is stillopen.
EISENBERG UNIQUENESS PAIR 17
We need the following two lemmas which are required to prove the necessarypart of Theorem 3.5. The main idea behind these lemmas is to pull down aninterval from some of the partitioning sets of the projection Π(Λ) . The aboveargument helps to negate the assumption that e Π(Λ) is not dense in R . Lemma 3.7.
Suppose I is an interval such that I ∩ Π ∗ (Λ) is dense in I. Thenthere exists an interval I ′ ⊂ I such that I ′ ⊂ S j =2 Π j (Λ) . Proof.
If ¯ ξ ∈ I ∩ Π ∗ (Λ) , then δ j ∈ L Π (Λ) , ¯ ξloc ; j = 0 , . By hypothesis, I ∩ Π ∗ (Λ)is dense in I, therefore there exists an interval I ¯ ξ ⊂ I containing ¯ ξ such that δ j can be extended continuously on I ¯ ξ . In addition, δ satisfies(3.11) | δ ( ¯ ξ ) | = (cid:12)(cid:12) e πi ¯ η + e πi ¯ η (cid:12)(cid:12) < , whenever ¯ ξ ∈ I ∩ Π ∗ (Λ) . Since δ is continuous on I ¯ ξ , we can extract aninterval I ′ ⊂ I ¯ ξ containing ¯ ξ such that | δ ( ξ ) | < ξ ∈ I ′ . Consequently, I ′ ∩ Π ∗ (Λ) is dense in I ′ . Now for ξ ∈ I ′ , there exists asequence ξ n ∈ I ′ ∩ Π ∗ (Λ) such that ξ n → ξ. Hence the corresponding imagesequences η ( n ) j ∈ Σ ξ n ⊆ [0 ,
2) will have convergent subsequences, say η ( n k ) j which converge to η j ; j = 0 , . Since the set Λ is closed, ( ξ, η j ) ∈ Λ for j = 0 , . Now, we only need to show that η = η . If possible, suppose η = η , then by the continuity of δ on I ′ , it follows that | δ ( ξ n ) | → | δ ( ξ ) | . However, | δ ( ξ n k ) | = (cid:12)(cid:12)(cid:12) e πiη ( nk )0 + e πiη ( nk )1 (cid:12)(cid:12)(cid:12) → . That is, | δ ( ξ ) | = 2 , which contradicts the fact that | δ ( ξ ) | < ξ ∈ I ′ . Thus, we infer that I ′ ⊂ S j =2 Π j (Λ) . (cid:3) Lemma 3.8.
Let I be an interval such that I ∩ Π ∗ (Λ) is dense in I. Thenthere exists an interval I ′ ⊂ I such that I ′ is contained in Π ∗ (Λ) ∪ Π (Λ) . Proof.
Let ¯ ξ ∈ I ∩ Π ∗ (Λ) , then e j ∈ L Π (Λ) , ¯ ξloc ; j = 0 , , . For p = 3 , Equation(3.10) yields ( e , e , e ) = ( − abc, ( ab + bc + ca ) , − ( a + b + c )) , where ( a, b, c ) = ( χ , χ , χ ) . Hence e j ; j = 0 , , ρ on Π (Λ) by ρ = (cid:0) a ( b − c ) + b ( c − a ) + c ( a − b ) (cid:1) . Since ρ is a symmetric polynomial in a, b, c, by the fundamental theorem ofsymmetric polynomials, ρ can be expressed as a polynomial in e j ; j = 0 , , . Moreover, ρ ( ¯ ξ ) = 0 . Hence it follows that ρ ∈ L Π (Λ) , ¯ ξloc . By hypothesis, I ∩ Π ∗ (Λ) is dense in I, there exists an interval I ¯ ξ ⊂ I containing ¯ ξ such that ρ can be continuously extended on I ¯ ξ . Thus, by continuity of ρ on I ¯ ξ , thereexists an interval J ⊂ I ¯ ξ containing ¯ ξ such that ρ ( ξ ) = 0 for all ξ ∈ J. Consequently, J ∩ Π ∗ (Λ) is dense in J and hence for ξ ∈ J, there exists asequence ξ n ∈ J ∩ Π ∗ (Λ) such that ξ n → ξ. Thus, the corresponding imagesequences η ( n ) j ∈ Σ ξ n ⊆ [0 ,
2) will have convergent subsequences, say η ( n k ) j which converge to η j ; j = 0 , ,
2. Since the set Λ is closed, ( ξ, η j ) ∈ Λ for j = 0 , , . Next, we claim that all of η j ; j = 0 , , ρ on J, it follows that ρ ( ξ ) = 0 , which contradicts the fact that ρ ( ξ ) = 0for all ξ ∈ J. Hence we infer that J ⊂ S j =3 Π j (Λ) . Further, using the facts that e j ∈ L Π (Λ) , ¯ ξloc and J ∩ Π ∗ (Λ) is dense in J, e j can be extended continuously onan interval I ′ ⊂ J containing ¯ ξ such that e j ( ξ ) = 0 for all ξ ∈ I ′ . That is, if ξ ∈ I ′ ∩ Π (Λ) , then e j ∈ L Π (Λ) ,ξloc and hence ξ ∈ Π ∗ (Λ) . Thus, we concludethat I ′ ⊂ Π ∗ (Λ) ∪ Π (Λ) . (cid:3) Proof of Theorem 3.5.
We first prove the sufficient part of Theorem 3.5.Suppose the set f Π p (Λ) is dense in R . Then we show that (Γ , Λ) is a Heisenberguniqueness pair. For ˆ µ | Λ = 0 , we claim that ˆ f k | f Π p (Λ) = 0 , whenever k ∈ F o . Since ˆ f k is a continuous function which vanishes on a dense set f Π p (Λ) , it followsthat ˆ f k ≡ k ∈ F o . Thus, µ = 0 . As the projection Π(Λ) is decomposed into the four pieces, the proof of theabove assertion will be carried out in the following four cases.( S ) . If ξ ∈ Π (Λ) , then there exist at least four distinct η j ∈ Σ ξ such thatˆ µ ( ξ, η j ) = 0 for all j ∈ F o . Hence ˆ f k ( ξ ); k ∈ F o satisfy a homogeneous systemof four equations. As ξ ∈ Π (Λ) , by using the property that H p − ( a , a , a ) = H p − ( a , a , a ) , we infer that ˆ f k ( ξ ) = 0 for all k ∈ F o . ( S ) . If ξ ∈ Π (Λ) , then there exist at least three distinct η j ∈ Σ ξ which satisfyˆ µ ( ξ, η j ) = 0; j = 0 , , . If ˆ f ( ξ ) = 0 , then we get ˆ f k ( ξ ) = 0 for k = 0 , , . Onthe other hand if ˆ f ( ξ ) = 0 , then we can substitute(3.12) ˆ f j ( ξ ) = τ j ( ξ ) ˆ f ( ξ ) , where τ j are defined on Π (Λ) for j = 0 , , . Hence X ξ = ( τ ( ξ ) , τ ( ξ ) , τ ( ξ ))will satisfy the system of equations A ξ X ξ = B pξ . By applying the Wienerlemma to Equations (3.12), we infer that τ j ∈ L Π (Λ) ,ξloc ; j = 0 , , . That is, ξ ∈ Π p ∗ (Λ) . Thus for ξ ∈ Π (Λ) r Π p ∗ (Λ) , we conclude that ˆ f k ( ξ ) = 0 for all k ∈ F o . EISENBERG UNIQUENESS PAIR 19 ( S ) . If ξ ∈ Π (Λ) , then there exist two distinct η j ∈ Σ ξ for which ˆ µ ( ξ, η j ) = 0 , whenever j = 0 , . That is,(3.13) ˆ f ( ξ ) + χ j ( ξ ) ˆ f ( ξ ) + χ j ( ξ ) ˆ f ( ξ ) + χ pj ( ξ ) ˆ f ( ξ ) = 0 , where χ j ( ξ ) = e πiη j ; j = 0 , . If ˆ f ( ξ ) = 0 , then by applying the Wiener lemmato Equations (3.13), it follows that ξ ∈ Π p ∗ (Λ) . That is, if ξ ∈ Π (Λ) r Π p ∗ (Λ) , then ˆ f ( ξ ) = 0 . Further, if ˆ f ( ξ ) = 0 and ˆ f ( ξ ) = 0 , then an application of the Wiener lemmato Equations (3.13), it follows that ξ ∈ Π ∗ (Λ) . By Lemma 3.3, ξ ∈ Π p ∗ (Λ) . Thus for ξ ∈ Π (Λ) r Π p ∗ (Λ) , we infer that ˆ f k ( ξ ) = 0 for all k ∈ F o . ( S ) . If ξ ∈ Π (Λ) , then there exists a unique η ∈ Σ ξ for which ˆ µ ( ξ, η ) = 0 . That is,(3.14) ˆ f ( ξ ) + χ ( ξ ) ˆ f ( ξ ) + χ ( ξ ) ˆ f ( ξ ) + χ p ( ξ ) ˆ f ( ξ ) = 0 , where χ ( ξ ) = e πiη . If ˆ f ( ξ ) = 0 , then by applying the Wiener lemma toEquation (3.14), it implies that χ ∈ P ,p [ L Π (Λ) ,ξloc ] . That is, ξ ∈ Π ∗ (Λ) . Thusfor ξ ∈ Π (Λ) r Π ∗ (Λ) , we have ˆ f ( ξ ) = 0 . Further, if ˆ f ( ξ ) = 0 and ˆ f ( ξ ) = 0 , then an application of the Wienerlemma to Equation (3.14), yields χ ∈ P , [ L Π (Λ) ,ξloc ] . By Lemma 3.2, it followsthat ξ ∈ Π ∗ (Λ) . That is, if ξ ∈ Π (Λ) r Π ∗ (Λ) , then ˆ f k ( ξ ) = 0 for k = 2 , . Finally, if ˆ f k ( ξ ) = 0 for k = 2 , f ( ξ ) = 0 , then by applying the Wienerlemma to Equation (3.14), we infer that χ ∈ L Π (Λ) ,ξloc . By Lemma 3.2, it followsthat ξ ∈ Π ∗ (Λ) . Thus for ξ ∈ Π (Λ) r Π ∗ (Λ) , we conclude that ˆ f k ( ξ ) = 0 forall k ∈ F o . Now, we prove the necessary part of Theorem 3.5. Suppose (Γ , Λ) is aHeisenberg uniqueness pair. Then we claim that the set e Π(Λ) is dense in R . We observe that this is possible if the dispensable sets Π j ∗ (Λ); j = 1 , , e Π(Λ) is not dense in R . Then there exists an openinterval I o ⊂ R such that I o ∩ e Π(Λ) is empty. This in turn implies that(3.15) Π(Λ) ∩ I o = [ j =1 Π j ∗ (Λ) ! ∩ I o . Thus from (3.15), it follows that I o intersects only the dispensable sets. Now,the remaining part of the proof of Theorem 3.5 is a consequence of the followingtwo lemmas which provide the interlacing property of the dispensable setsΠ j ∗ (Λ); j = 1 , , . Lemma 3.9.
There does not exist any interval J ⊂ I o such that Π(Λ) ∩ J iscontained in Π j ∗ (Λ); j = 1 , , . Proof.
On the contrary, suppose there exists an interval J ⊂ I o such thatΠ(Λ) ∩ J ⊂ Π j ∗ (Λ) , for some j ∈ { , , } . Since Π j ∗ (Λ); j = 1 , , a ) . If ξ ∈ Π(Λ) ∩ J ⊂ Π ∗ (Λ) , then χ ∈ P ,p [ L Π (Λ) ,ξloc ] . Hence there exists aninterval I ξ ⊂ J containing ξ and ϕ k ∈ L ( R ); k = 0 , , χ p + ˆ ϕ χ + ˆ ϕ χ + ˆ ϕ = 0on I ξ ∩ Π (Λ) . Now, consider a function f ∈ L ( R ) such that ˆ f ( ξ ) = 0 andsupp ˆ f ⊂ I ξ . Let f k = f ∗ ϕ k ; k = 0 , , . Then we can construct a Borelmeasure µ which is supported on Γ such thatˆ µ ( ξ, η ) = ˆ f ( ξ ) + χ ( ξ ) ˆ f ( ξ ) + χ ( ξ ) ˆ f ( ξ ) + χ p ( ξ ) ˆ f ( ξ ) = 0for all ξ ∈ I ξ ∩ Π ∗ (Λ) , where η ∈ Σ ξ . Since (3.15) yields I ξ ∩ Π(Λ) = I ξ ∩ Π ∗ (Λ) , it implies that ˆ µ | Λ = 0 . However, µ is a non-zero measure which contradictsthe fact that (Γ , Λ) is a HUP.( b ) . If ξ ∈ Π(Λ) ∩ J ⊂ Π ∗ (Λ) , then by Lemma 3.3, ξ ∈ Π p ∗ (Λ) . Hence thereexists an interval I ξ ⊂ J containing ξ and ϕ k ∈ L ( R ); k = 0 , , χ pj + ˆ ϕ χ j + ˆ ϕ χ j + ˆ ϕ = 0on I ξ ∩ Π (Λ) for j = 0 , . Let f ∈ L ( R ) be such that ˆ f ( ξ ) = 0 andsupp ˆ f ⊂ I ξ . Denote f k = f ∗ ϕ k ; k = 0 , , . Then we can construct a Borelmeasure µ that satisfiesˆ µ ( ¯ ξ, ¯ η j ) = ˆ f ( ¯ ξ ) + χ j ( ¯ ξ ) ˆ f ( ¯ ξ ) + χ j ( ¯ ξ ) ˆ f ( ¯ ξ ) + χ pj ( ¯ ξ ) ˆ f ( ¯ ξ ) = 0for all ¯ ξ ∈ I ξ ∩ Π ∗ (Λ) and j = 0 , . Since I ξ ∩ Π(Λ) = I ξ ∩ Π ∗ (Λ) , it followsthat ˆ µ | Λ = 0 , though µ is a non-zero measure.( c ) . If ξ ∈ Π(Λ) ∩ J ⊂ Π ∗ (Λ) , then by Lemma 3.4, it follows that ξ ∈ Π p ∗ (Λ) . As τ k ∈ L Π (Λ) ,ξloc ; k = 0 , , , there exists an interval I ξ ⊂ J containing ξ and ϕ k ∈ L ( R ) such that ˆ ϕ k = τ k on I ξ ∩ Π (Λ) for k = 0 , , . Let f ∈ L ( R ) besuch that ˆ f ( ξ ) = 0 and supp ˆ f ⊂ I ξ . Denote f k = f ∗ ϕ k ; k = 0 , , . Since X ξ = ( τ ( ξ ) , τ ( ξ ) , τ ( ξ )) satisfies A ξ X ξ = B pξ , we have τ + χ j τ + χ j τ + χ pj = 0on I ξ ∩ Π (Λ) for j = 0 , , . Hence we can construct a Borel measure µ suchthat ˆ µ ( ¯ ξ, ¯ η j ) = ˆ f ( ¯ ξ ) + χ j ( ¯ ξ ) ˆ f ( ¯ ξ ) + χ j ( ¯ ξ ) ˆ f ( ¯ ξ ) + χ pj ( ¯ ξ ) ˆ f ( ¯ ξ ) = 0for all ¯ ξ ∈ I ξ ∩ Π ∗ (Λ) and j = 0 , , . As I ξ ∩ Π(Λ) = I ξ ∩ Π ∗ (Λ) , we inferthat ˆ µ | Λ = 0 , even though µ is a non-zero measure. (cid:3) The next lemma is to deal with the situation that any interval J ⊂ I o cannot contain only the points of any pair of dispensable sets. Lemma 3.10.
There does not exist any interval J ⊂ I o such that Π(Λ) ∩ J iscontained in Π j ∗ (Λ) ∪ Π k ∗ (Λ) ∀ j = k and j, k ∈ { , , } . EISENBERG UNIQUENESS PAIR 21
Proof.
On the contrary, suppose there exists an interval J ⊂ I o such thatΠ(Λ) ∩ J ⊂ Π j ∗ (Λ) ∪ Π k ∗ (Λ) for some j = k and j, k ∈ { , , } . Then we havethe following three cases: (a).
If Π(Λ) ∩ J ⊂ Π ∗ (Λ) ∪ Π ∗ (Λ) , then Equation (3.15) yields(3.16) J ∩ Π(Λ) = J ∩ (cid:0) Π ∗ (Λ) ∪ Π ∗ (Λ) (cid:1) . We claim that J ∩ Π ∗ (Λ) is dense in J. If possible, suppose there exists aninterval I ⊂ J such that Π ∗ (Λ) ∩ I = ∅ . Then from (3.16), we get I ∩ Π(Λ) = I ∩ Π ∗ (Λ) ⊂ Π ∗ (Λ) which contradicts Lemma 3.9. By Lemma 3.7, there existsan interval I ′ ⊂ J such that I ′ ⊂ S j =2 Π j (Λ) . This contradicts the assumptionthat I o intersects only the dispensable sets. (b). If Π(Λ) ∩ J ⊂ Π ∗ (Λ) ∪ Π ∗ (Λ) , then J ∩ Π(Λ) = J ∩ (cid:0) Π ∗ (Λ) ∪ Π ∗ (Λ) (cid:1) . As similar to the case (a) , J ∩ Π ∗ (Λ) is also dense in J. Hence by Lemma 3.8,there exists an interval I ′ ⊂ J such that I ′ is contained in Π ∗ (Λ) ∪ Π (Λ) . Thusin view of Lemma 3.9, we have arrived at a contradiction to the assumptionthat I o intersects only the dispensable sets. (c). If Π(Λ) ∩ J ⊂ Π ∗ (Λ) ∪ Π ∗ (Λ) , then J ∩ Π(Λ) = J ∩ (Π ∗ (Λ) ∪ Π ∗ (Λ)) . Hence it follows that J ∩ Π ∗ (Λ) is dense in J. By using Lemma 3.8, thereexists an interval I ′ ⊂ J such that I ′ is contained in Π ∗ (Λ) ∪ Π (Λ) , whichcontradict the assumption that I o intersects only the dispensable sets. (cid:3) Finally, since Π(Λ) is a dense subset of R , in view of Lemmas 3.9 and 3.10,the only possibility that any interval J ⊂ I o would intersect all the dispensablesets Π j ∗ (Λ); j = 1 , , . We claim that Π ∗ (Λ) ∩ I o is dense in I o . Otherwise,there exists an interval I ⊂ I o such that Π ∗ (Λ) ∩ I = ∅ . Then from (3.15),we get I ∩ Π(Λ) ⊂ (cid:0) Π ∗ (Λ) ∪ Π ∗ (Λ) (cid:1) which contradicts Lemma 3.10. Henceby Lemma 3.7, there exists an interval I ′ ⊂ I o such that I ′ is contained inΠ (Λ) ∪ Π (Λ) ∪ Π (Λ) which contradicts the assumption that I o intersectsonly the dispensable sets. Concluding remarks:(a).
We observe a phenomenon of interlacing of three totally disconnecteddisjoint dispensable sets Π (3 − j ) ∗ (Λ) : j = 0 , , (b). If the measure in question is supported on an arbitrary number of parallellines, then the size of the dispensable sets would be larger. Indeed, the methodused for the proof of Theorem 3.5 would be highly implicit for a large number ofparallel lines. Since the dispensable sets are totally disconnected, it would bean interesting question to analyze Heisenberg uniqueness pairs correspondingto the finite number of parallel lines in terms of Hausdorff dimension of thedispensable sets. (c).
If we consider countably many parallel lines, then whether the projectionΠ(Λ) would be still relevant after deleting the countably many dispensablesets, seems to be a reasonable question. We leave these questions open for thetime being. (d).
For p = 3 , in Lemma 3.8 we have used the fact that any symmetricpolynomial in a, b, c can be expressed as a polynomial in τ j ; j = 0 , , . Thisenables us to define a function ρ ∈ L Π (Λ) , ¯ ξloc , which is crucial in the proof ofLemma 3.8. However, for p ≥ , the functions τ j ; j = 0 , , τ j ; j = 0 , , , then we can think to modify the Lemma 3.8 in terms of Π p ∗ (Λ) that wouldhelp in minimizing the size of the set e Π(Λ) . Hence a characterization of Λ forfour lines problem might be obtained that would be closed to three lines result.However, an exact analogue of three lines result for a large number of lines isstill open.
Acknowledgements:
The authors would like to thank the referee for his/her fruitful suggestion.The authors wish to thank E. K. Narayanan and Rama Rawat for severalfruitful discussions during preparation of this manuscript. The authors wouldalso like to gratefully acknowledge the support provided by IIT Guwahati,Government of India.
References [1] M. Benedicks,
On Fourier transforms of functions supported on sets of finite Lebesguemeasure,
J. Math. Anal. Appl. 106 (1985), no.1, 180-183.[2] D. B. Babot,
Heisenberg uniqueness pairs in the plane, Three parallel lines,
Proc.Amer. Math. Soc. 141 (2013), no. 11, 3899-3904.[3] S. C. Bagchi and A. Sitaram,
Determining sets for measures on R n , Illinois J. Math.26 (1982), no. 3, 419-422.[4] W. F. Donoghue, Jr.,
Distribution and Fourier transforms,
Academic Press, NewYork, 1969.[5] L. Flatto, D. J. Newman, and H. S. Shapiro,
The level curves of harmonic polyno-mials,
Trans. Amer. Math. Soc. 123 (1966), 425-436.[6] V. Havin, and B. Jricke,
The Uncertainty Principle in Harmonic Analysis,
Ergebnisseder Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and RelatedAreas (3)], 28. Springer-Verlag, Berlin, 1994.[7] H. Hedenmalm, A. Montes-Rodrguez,
Heisenberg uniqueness pairs and the Klein-Gordon equation,
Ann. of Math. (2) 173 (2011), no. 3, 1507-1527.[8] P. Jaming and K. Kellay,
A dynamical system approach to Heisenberg uniquenesspairs,
J. Anal. Math. (appearing) arXiv:1312.6236, July, 2014.[9] J.P. Kahane,
Sries de Fourier absolument convergentes,
Band 50 Springer-Verlag,Berlin-New York 1970.[10] N. Lev,
Uniqueness theorem for Fourier transform. Bull. Sc. Math.,
EISENBERG UNIQUENESS PAIR 23 [11] I.G. Macdonald,
Symmetric Functions and Hall Polynomials,
Second ed., OxfordUniversity Press, New York, 1995.[12] E. K. Narayanan and P. K. Ratnakumar,
Benedick’s theorem for the Heisenberggroup,
Proc. Amer. Math. Soc. 138 (2010), no. 6, 2135-2140.[13] V. Pati and A. Sitaram,
Some questions on integral geometry on Riemannian man-ifolds,
Ergodic theory and harmonic analysis (Mumbai, 1999). Sankhya Ser. A 62(2000), no. 3, 419-424.[14] R. Rawat, A. Sitaram,
Injectivity sets for spherical means on R n and on symmetricspaces, Jour. Four. Anal. Appl., 6 (2000), 343-348.[15] F. J. Price and A. Sitaram,
Functions and their Fourier transforms with supports offinite measure for certain locally compact groups,
J. Funct. Anal. 79 (1988), no. 1,166-182.[16] P. Sjolin,
Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin,
Bull.Sc. Math., 135(2011), 125-133.[17] P. Sjolin,
Heisenberg uniqueness pairs for the parabola,
Jour. Four. Anal. Appl.,19(2013), 410-416.[18] A. Sitaram, M. Sundari and S. Thangavelu,
Uncertainty principles on certain Liegroups,
Proc. Indian Acad. Sci. Math. Sci. 105 (1995), no. 2, 135-151.[19] F. J. Gonzalez Vieli,
A uniqueness result for the Fourier transform of measures onthe sphere,
Bull. Aust. Math. Soc. 86(2012), 78-82.[20] F. J. Gonzalez Vieli,
A uniqueness result for the Fourier transform of measures onthe paraboloid,
Matematicki Vesnik, Vol. 67(2015), No. 1, 52-55.
Department of Mathematics, Indian Institute of Technology, Guwahati,India 781039.
E-mail address ::