Restriction of toral eigenfunctions to hypersurfaces and nodal sets
Let be a smooth Riemannian surface without boundary, the corresponding Laplace-Beltrami operator and a smooth curve in . Burq, Gérard and Tzvetkov [BGT] established bounds for the -norm of the restriction of eigenfunctions of to the curve , showing that if , , then
and if has non-vanishing geodesic curvature then (1.1) may be improved to
due to the fact that there is a corresponding bound on the supremum of the eigenfunctions. They raise the question whether in (1.3) the factor can be replaced by a constant, that is whether there is a uniform restriction bound. As pointed out by Sarnak [Sar2], if we take to be a geodesic segment on the torus, this particular problem is essentially equivalent to the currently open question of whether on the circle , the number of lattice points on an arc of size admits a uniform bound.
In [BGT] results similar to (1.1) are also established in the higher dimensional case for restrictions of eigenfunctions to smooth submanifolds, in particular (1.1) holds for codimension-one submanifolds (hypersurfaces) and is sharp for the sphere . Moreover (1.2) remains valid for hypersurfaces with positive curvature [H].
In this paper we pursue the improvements of (1.2) for the standard flat -dimensional tori , considering the restriction to (codimension-one) hypersurfaces with non-vanishing curvature.
Main Theorem. Let and let be a real analytic hypersurface with non-zero curvature. There are constants and , all depending on , so that all eigenfunctions of the Laplacian on with satisfy
Observe that for the lower bound, the curvature assumption is necessary, since the eigenfunctions all vanish on the hypersurface . In fact this lower bound implies that a curved hypersurface cannot be contained in the nodal set of eigenfunctions with arbitrarily large eigenvalues.
It was shown in [B-R1] that this last property of the nodal sets of toral eigenfunctions hold in arbitrary dimension . As we point out in Section 10, the argument from [B-R1] implies in fact a bound for the dimensional Hausdorff measure of the intersection of nodal sets with a fixed hypersurface :
Let be a real analytic hypersurface with nowhere vanishing curvature. Then for , the nodal set of any eigenfunction satisfies
For dimension , this means an upper bound for the number of intersection points of a fixed curve with the nodal lines. Interestingly, using the Main Theorem, one can show that conversely:
Let be a real analytic non-geodesic curve. There is such that for , the nodal set of any eigenfunction satisfies
and for , the following property
Let be as in the Main Theorem. There is such that for , the nodal set of any eigenfunction intersects .
Returning to the results of [BGT] for smooth Riemannian surfaces, let us point out that there is a close connection between estimates on with a geodesic segment and bounds on the -norm . Recall Sogge’s general estimate for the -norm [So1]
The following inequalities were established in [B]
if is a geodesic segment and , and conversely
where the maximum is over all geodesic segments of unit length. Hence (1.9), (1.10) imply that improving upon the restriction bound (1.1) is essentially equivalent with convexity breaking for the -norm (see also [So2]). Of course for and previous considerations are of no interest. However, the example of an integrable torus constructed in [B2] provides a sequence of eigenfunctions and a geodesic segment such that
The proof of the Main Theorem for is rather simple (compared with ) and we describe it next, as an illustration of the method and some of the arithmetic ingredients used, see [BR].
Denote by the normalized arc-length measure on the curve . Using the method of stationary phase, one sees that if has non-vanishing curvature then the Fourier transform decays as
Moreover with equality only for , hence
for some .
An eigenfunction of the Laplacian on is a trigonometric polynomial of the form:
(where ), all of whose frequencies lie in the set . As is well known, in dimension , for all . Moreover, by a result of Jarnik [J], any arc on of length at most contains at most two lattice points (Cilleruelo and Cordoba [CC] showed that for any , arcs of length contain at most lattice points and in [CG] it is conjectured that this remains true for any ). Hence we may partition,
where and for . Correspondingly we may write,
so that and
Applying (1.12) we see that if and because the total sum of these nondiagonal terms is bounded by . It suffices then to show that the diagonal terms satisfy
This is clear if while if then
The proof of the Main Theorem for dimension is considerably more involved and occupies Sections 2–9 of the paper. Arguing along the lines of the two-dimensional case gives an upper bound of . To get the uniform bound for we need to replace the upper bound (1.12) for the Fourier transform of the hypersurface measure by an asymptotic expansion, and then exploit cancellation in the resulting exponential sums over the sphere. A key ingredient there is controlling the number of lattice points in spherical caps.
To state some relevant results, denote as before by the set of lattice points on the sphere of radius . We have . Let be the maximal number of lattice points in the intersection of with a spherical cap of size . A higher-dimensional analogue of Jarnik’s theorem implies that if then all lattice points in such a cap are co-planar, hence in that case, for any . For larger caps, we show:
i) Let . Then for any ,
ii) Let . Then
iii) For we have
(the factor is redundant for large ).
The term concerns the equidistribution of , while the term measures deviations related to accumulation in lower dimensional strata.
Only (1.19) is relevant for our purpose (Lemma 6.8 in the paper, proved in Section 9) and (1.20), (1.21) for (proven in Appendix A) were included to provide a more complete picture. We point out that the argument used to obtain (1.19) is based on certain diophantine considerations and dimension reduction, hence differs considerably from the proof of (1.20), (1.21) using standard Hardy-Littlewood circle method and Kloosterman’s refinement for .
The second result expresses a mean-equidistribution property of . Partition the sphere into sets of size , for instance by intersecting with cubes of that size. Since , one may expect that . We show (in joint work with P. Sarnak [BRS]) that as a consequence of “Linnik’s basic Lemma”, this holds in the mean square:
Finally, considering very large caps , there is an estimate
some absolute constant)
Let and let , be spherical -caps on of mutual distance at least . Following the argument for , we need to bound exponential sums of the form
where is the support function of the hyper-surface , which appears in the asymptotic expansion of the Fourier transform of the surface measure on , see Section 3. For instance, in the case that is the unit sphere then .
There is so that (1.24) admits a bound of for .
This statement depends essentially on the equidistribution of in caps of size , as expressed in Lemma 1.5.
Using Taylor expansions of the function with restricted to and suitable coordinate restrictions, Lemma 1.7 is eventually reduced to the following one-dimensional exponential sum estimate (proven in Section 6):
Let and arbitrary discrete sets such that for and . Then
for some .
Extending the Main Theorem to arbitrary dimension remains unsettled at this point. We make the following
Let be arbitrary and a real analytic hypersurface. Then, for some constant , all eigenfunctions of satisfy
If moreover has nowhere vanishing curvature and , for some , also
It should be pointed out that in our proof of the Main Theorem for , only distributional properties of were exploited, but not the fact that actually consists of lattice points. In Section 11, we give an example, for , of sets satisfying the ‘ideal’ distributional property
and such that the Fourier restriction operator
has unbounded norm for . This illustrates the difficulty for carrying out our analysis in larger dimension.
As said earlier, even for and a straight line segment in , (1.26) remains open and is roughly equivalent with the arithmetic statement that the number of lattice points on an arc of size on the circle is bounded by an absolute constant. An easy argument in [BR2] shows that this last property is true for most and in fact the elements of are at least separated, for all . In Section 12, we establish the following
Let be a smooth curve. Then for almost all , there is a uniform restriction bound
Let and be sufficiently large. Then for a ‘typical’ element of the eigenfunction space , the nodal set has components.
Recalling Courant’s nodal domain theorem, the interest of Theorem 1.11 is the lower bound on the number of nodal domains.
Almost all the subsequent analysis in the paper relates to and -eigenfunctions. Let us stress again that the arithmetic structure of the frequencies of the trigonometric polynomials involved is essential here.
Acknowledgement: The authors are indebted to P. Sarnak for many stimulating discussions on the material presented in the paper. J.B. was supported in part by N.S.F. grant DMS 0808042. Z.R. was supported by the Oswald Veblen Fund during his stay at the Institute for Advanced Study and by the Israel Science Foundation (grant No. 1083/10).
2. Lattice Points in Spherical Caps
2.1. Lattice points on spheres
We recall what is known concerning the total number of lattice points on the sphere of radius . Throughout we assume, as we may, that is an integer. We have a general upper bound
and in dimension we in fact have both a lower and upper bound of this strength:
In smaller dimensions both the lower and upper bound (2.1) need not hold. For instance if is a power of 2 then is bounded. The situation in dimension is particularly delicate. It is known that if and only if . There are primitive lattice points on the sphere of radius (that is with ) if an only if . Concerning the number of lattice points, the upper bound (2.1) is still valid, and if there are primitive lattice points then there is a lower bound of but there are arbitrarily large ’s so that
A fundamental result conjectured by Linnik (and proved by him assuming the Generalized Riemann Hypothesis), that for , the projections of these lattice points to the unit sphere become uniformly distributed on the unit sphere as . This was proved unconditionally by Duke [D, D-SP] and Golubeva and Fomenko [G-F], following a breakthrough by Iwaniec [I].
2.2. Lattice points in spherical caps: Statement of results
Let be a unit vector, , and . Consider the spherical cap which is the intersection of the sphere with the ball of radius around . Set
which is the maximal number of lattice points in a spherical cap of size on the sphere . We want to give an upper bound for in the case of dimension . The results which will be proven in this section are as follows:
i) For all ,
This is an immediate consequence of a Jarnik-type result on non-coplanar lattice points in small caps. It is only useful for small caps, when .
For larger caps we shall show the following bound:
ii) For any ,
It is natural to conjecture that for .
2.3. Intersections with hyperplanes
Let be the maximal number of lattice points in the intersection of the sphere in and a hyperplane.
For dimension ,
while in dimension we have
2.4. Small caps
For a spherical cap of size on the sphere of radius in the number of lattice points in is at most
Firstly, we note that if the cap has radius then it contains only a lattice points. This can be deduced from Jarnik’s method [J] and also from a general result of Andrews [A] that if is any convex body in with volume then the number of lattice points on its boundary which are not coplanar is . In our case of a cap in dimension 3, the base of the cap has area and if is the opening of the cap, so that , then the height of the cap is about , hence the volume of the cap is . Thus if then any such cap will contain at most (say) 100 non-coplanar lattice points. Any lattice points in the cap will lie on one of the plane sections of the cap through any three of the 100 non-coplanar lattice points. Each such plane section will contain at most lattice points (uniformly as a function of the plane) and hence the cap will contain at most lattice points.
Now, for a cap of radius , divide it into caps of radius ; the number of such caps will be area, and hence the total number of lattice points in is at most . ∎
2.5. A linear and sub-linear bound
We now turn to larger caps.
Here is a simple bound via slicing, using the fact that we can control the number of lattice points in the intersection of a sphere and a hyperplane parallel to one of the coordinate hyperplanes:
In dimension ,
Proof. A ball of radius is contained in a vertical slab of the form and hence all integer points in the intersection of the sphere and the ball lie in the union of the planes , with integer. The intersection of each plane and the sphere has at most lattice points, and therefore the total number of lattice points is at most .
In particular, for dimension this says that
We can improve on Lemma 2.2 by slicing with well-chosen planes rather than vertical planes. More precisely, we have
Let be a cap of size on the sphere . Then for any ,
It will involve several considerations.
i) Finding good slices. We try to find an integer vector and use slices of the cap with the sections . We consider a larger cap of radius around which contains the original cap . Thus we want the new cap angle to satisfy
To bound the number of lattice points in the new cap , we exhaust them by the parallel sections , which are orthogonal to the direction of the new cap. The distance between adjacent sections is . The number of sections intersecting the cap is bounded by times the height of the cap, which is . Hence the number of sections intersecting the cap is
and the analysis above shows that the number of lattice points in the cap is bounded by
To gain over the linear bound (2.9) we need to find some and a nonzero integer vector such that
then (2.15) is implied by requiring both
where we have set
Finding as above is then our goal.
(ii) Diophantine approximation
Fix an integer and . Let . Then one of the following holds:
(1) There is and such that
(2) There are , with with
where we denote by the fractional part of , or the distance of to the nearest integer.
Proof. Let be a smooth bump function on the torus , such that
If (2.20) fails, then
Expressing this in a Fourier series gives (writing
Hence for some nonzero .
Let , and an integer.
Then there is an integer and so that
Remark. Dirichlet’s principle says that given , and an integer , we can find and so that
Lemma 2.5 improves on this when is small.
Proof. Applying Lemma 2.4 with , either we have an integer with
which gives us what we need, or else the second option in the statement of the lemma occurs, that is there is some nonzero vector with , and so that
Now choose an integer so that
which is possible if and .
Using Dirichlet’s principle, there is some and an integer so that
We claim that these satisfy the statement of the Lemma. Indeed, by (2.28) we have
and due to (2.27) we have, since , that
giving . Moreover using the small linear relation (2.26) between and and replacing by we find
Now since we have
and combining with (2.31) we get
2.6. Proof of Lemma 2.3
3. The Fourier transform of surface-carried measures
Let be a real analytic surface with non-vanishing curvature and . Applying a rigid motion, we may assume and locally parametrized around by a map
where is real-analytic on a neighborhood of as has the form
Distinguishing the case (positive curvature) and (negative curvature), we need to consider the two models