Comptes Rendus Mathématique

Radu Ignat

Comptes Rendus
Mathématique
Radu Ignat
A short proof of the C 1,1 regularity for the eikonal equation
Volume 363 (2025), p. 887-891
Online since: 21 July 2025
https://doi.org/10.5802/crmath.768
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 887-891
https://doi.org/10.5802/crmath.768
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
A short proof of the C 1,1 regularity for the
eikonal equation
Une preuve simple de la régularité C 1,1 pour l’équation
eikonale
Radu Ignat a
a Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT,
31062 Toulouse Cedex 9, France
E-mail: radu.ignat@math.univ-toulouse.fr
In the memory of my PhD advisor Haïm Brezis
Abstract. We give a short and self-contained proof of the interior C 1,1 regularity of solutions ϕ : Ω→R to the
eikonal equation |∇ϕ| = 1 in an open setΩ⊂RN in dimension N ≥ 1 under the assumption thatϕ is pointwise
differentiable inΩ.
Résumé. Nous présentons une preuve courte et auto-contenue de la régularité intérieure C 1,1 des solutions
ϕ : Ω→R de l’équation eikonale |∇ϕ| = 1 dans un ouvertΩ⊂RN en toute dimension N ≥ 1 sous l’hypothèse
que ϕ est différentiable en tout point deΩ.
Keywords. Eikonal equation, characteristics, regularity.
Mots-clés. Équation eikonale, caractéristiques, régularité.
2020 Mathematics Subject Classification. 35F21, 35B65.
Funding. The author is partially supported by the ANR projects ANR-21-CE40-0004 and ANR-22-CE40-0006-
01.
Note. Article submitted by invitation.
Note. Article soumis sur invitation.
Manuscript received 1 February 2025, revised 3 June 2025, accepted 10 June 2025.
1. Introduction
Haïm Brezis always liked short and new proofs of known results. The aim of this note is to give
such a short and self-contained proof of the following result known in the theory of Hamilton–
Jacobi equations.
Theorem 1. Let Ω ⊂ RN be an open set in dimension N ≥ 1 and ϕ : Ω → R be a pointwise
differentiable solution to the eikonal equation |∇ϕ| = 1 inΩ. Then ∇ϕ is locally Lipschitz inΩ.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
888 Radu Ignat
The usual (standard) proof of this result is based on the following steps (see e.g. Lions [7],
Cannarsa–Sinestrari [2]): first, one checks that ϕ (and −ϕ) is a viscosity solution to the eikonal
equation (see [2, Definition 5.2.1]); second, one proves that ϕ is both semiconcave and semicon-
vex with linear modulus (see [2, Theorem 5.3.7]); third, one proves that ϕ is C 1 (see [2, Theo-
rem 3.3.7]) and finally, that ϕ is locally C 1,1 inΩ (see [2, Corollary 3.3.8]).
Our approach is based on the geometry of characteristics associated to the eikonal equation.
More precisely, if x0 ∈Ω, we say that X := Xx0 is a characteristic of a solution ϕ passing through
x0 in some time interval t ∈ [−T,T ] if{
Ẋ (t ) =∇ϕ(X (t )) for t ∈ [−T,T ],
X (0) = x0.
(1)
Then the beautiful proof of Caffarelli–Crandall [1, Lemma 2.2] shows in a short and self-contained
manner that every point x0 ∈Ω has a characteristic Xx0 that is a straight line along which ∇ϕ is
constant andϕ is affine. Finally, we give a geometric argument on the structure of characteristics
yielding the locally Lipschitz regularity of ∇ϕ inΩ.
The regularity result in Theorem 1 is optimal: such solution ϕ of the eikonal equation is not
C 2 in general (see e.g. [5, Proposition 1]). We mention that a more general regularizing effect (i.e.,
∇ϕ is locally Lipschitz away from vortex point singularities) is proved under a weaker assumption
∇ϕ ∈ W 1/p,p for p ∈ [1,3], see [3,5]. Similar results are obtained in the context of the Aviles–Giga
model which can be seen as a regularization of the eikonal equation (see [4,6]).
2. Proof of the main result
The first step is to show that each point x0 ∈Ω has a characteristic X := Xx0 that is a straight line
in direction ∇ϕ(x0). Moreover, ∇ϕ is constant while ϕ is affine along this characteristic. This fact
yields ϕ ∈ C 1(Ω). In order to have a self-contained proof of Theorem 1, we repeat here the very
nice argument of Caffarelli–Crandall [1, Lemma 2.2] based on a maximum type principle for the
eikonal equation.
Lemma 2. Let Ω ⊂ RN be an open set and ϕ : Ω→ R be a pointwise differentiable solution of the
eikonal equation |∇ϕ| = 1 inΩ. Then for every x0 ∈Ω, X (t ) = x0 + t∇ϕ(x0) is a characteristic of (1)
and
∇ϕ(X (t ))=∇ϕ(x0), ϕ(X (t ))=ϕ(x0)+ t , ∀ t ∈ [−T,T ],
for some T > 0. As a consequence, ϕ ∈C 1(Ω).
Proof. This proof follows the lines of [1, Lemma 2.2]. Let R > 0 be such that B R (x0) ⊂ Ω and
consider
Mr = max
B r (x0)
ϕ, mr = min
B r (x0)
ϕ, ∀ r ∈ [0,R].
Claim 3. Mr =ϕ(x0)+ r and mr =ϕ(x0)− r for every r ∈ [0,R].
Proof. For r ∈ [0,R], we pick some maximum point x+r ∈ B r (x0) such that ϕ(x+r ) = Mr . First,
we show that r ∈ [0,R] 7→ Mr is a nondecreasing 1-Lipschitz function. Indeed, for R ≥ r > r̃ , as
|x+r −x0| ≤ r , we can find a vector e ∈RN such that |e| ≤ r−r̃ and |x+r +e−x0| ≤ r̃ , i.e., x+r +e ∈ B r̃ (x0);
this yields
0 ≤ Mr −Mr̃ ≤ϕ(x+r )−ϕ(x+r +e) ≤ |e| ≤ r − r̃
because ϕ is 1-Lipschitz. Second, we prove that dMrdr = 1 a.e. in (0,R) because for r ∈ (0,R) and for
small h > 0, as x+r +h∇ϕ(x+r ) ∈ B r+h(x0), we have
liminf
h→0
Mr+h −Mr
h
≥ liminf
h→0
ϕ
(
x+r +h∇ϕ(x+r )
)−ϕ(x+r )
h
= ∣∣∇ϕ(x+r )∣∣2 = 1.
Radu Ignat 889
As M0 =ϕ(x0), we conclude Mr =ϕ(x0)+ r . Up to changing ϕ in −ϕ, one also gets mr =ϕ(x0)− r
for r ∈ [0,R]. □
To conclude the proof of Lemma 2, pick some minimum point x−R ∈ B R (x0) such that ϕ(x−R ) =
mR =ϕ(x0)−R (by Claim 3). As ϕ is 1-Lipschitz, we have, again by Claim 3:
2R =ϕ(x+R )−ϕ(x−R ) ≤ |x+R −x−R | ≤ 2R,
which means that [x+R , x
−
R ] is a diameter in B R (x0). Note that x
+
R (resp. x
−
R ) is the unique maximum
(resp. minimum) of ϕ in B R (x0) because if x̃+R is another maximum, then it has to be antipodal
to x−R , that is, x̃
+
R = x+R (the same for the uniqueness of x−R ). In particular, e∗ =
x+R −x0
R ∈ SN−1.
Define g : [−R,R] → R by g (r ) = ϕ(x0 + r e∗)−ϕ(x0). Then g is 1-Lipschitz and g (±R) = ϕ(x±R )−
ϕ(x0) = ±R (by Claim 3). So g (r ) = r for every r ∈ (−R,R) yielding 1 = g ′(r ) = e∗ · ∇ϕ(x0 + r e∗)
for every r . Thus, ∇ϕ(x0 + r e∗) = e∗ for every r ∈ [−R,R], in particular, e∗ = ∇ϕ(x0), i.e.,
X (r ) = x0 + r∇ϕ(x0) is a characteristic of (1) and
∇ϕ(X (r ))=∇ϕ(x0), ϕ(x0 + r∇ϕ(x0))=ϕ(x0)+ r, ∀ r ∈ [−R,R].
In particular, the (unique) maximum and minimum of ϕ in B R (x0) are achieved at the points
x±R = x0 ±R∇ϕ(x0).
It remains to prove that ∇ϕ is continuous in Ω. Indeed, let xn → x0 in Ω and B R (xn) ⊂Ω for
large n. Up to a subsequence, we may assume that ∇ϕ(xn) → e ∈SN−1. By above, we know that
ϕ
(
xn +R∇ϕ(xn)
)=ϕ(xn)+R. Passing to the limit, we obtainϕ(x0+Re) =ϕ(x0)+R, meaning that
x0 +Re is the maximum of ϕ in B R (x0). By uniqueness of the maximum point x+R , we conclude
that e = ∇ϕ(x0). The uniqueness of the limit e for such subsequences yield the convergence of
the whole sequence
(∇ϕ(xn))n to ∇ϕ(x0). □
Proof of Theorem 1. Let B be a ball, B ⊂Ω and we consider d ∈ (0, dist(B ,∂Ω)5 ). We will prove the
following.
Claim 4. There exists a universal constant C > 0 such that∣∣∇ϕ(x)−∇ϕ(y)∣∣É C
d
|x − y |, ∀ x, y ∈ B with |x − y | < d
10
.
Proof. Let Xx and X y be the characteristics passing through x and y constructed in Lemma 2
(that are lines in direction ∇ϕ(x) and ∇ϕ(y)). If Xx and X y coincide inside B (in particular,
∇ϕ(x) =±∇ϕ(y) by Lemma 2), then Lemma 2 implies that ∇ϕ(x) =∇ϕ(y) and the claim is trivial
in that case. Otherwise, ∇ϕ(x) ̸= ±∇ϕ(y) and Xx and X y cannot intersect inside Ω (as ∇ϕ is
continuous inΩ by Lemma 2). Let |x − y | < d10 and x ′ be the projection of y on Xx . Clearly,
ℓ= |x ′− y | ≤ |x − y | < d
10
,
dist(x ′,∂Ω) > 5d −ℓ and ∇ϕ(x ′) = ∇ϕ(x). Up to changing ϕ in −ϕ, we may assume that ϕ(x ′) ≤
ϕ(y) and up to an additive constant, we also may assume that 0 = ϕ(x ′) ≤ ϕ(y) = a. As |∇ϕ| = 1
on the segment [x ′y], we deduce that 0 ≤ a = ϕ(y)−ϕ(x ′) ≤ |x ′ − y | = ℓ. Let z be the point on
the characteristic X y reached at time t = d in direction ∇ϕ(y), i.e., |z − y | = d , ∇ϕ(y) = z−yd and
ϕ(z) = a +d ; in particular, z ∈ Ω. Let w be the point on the characteristic Xx′ reached at time
t = −d in direction ∇ϕ(x ′), i.e., |x ′−w | = d , ∇ϕ(x ′) = x′−wd and ϕ(w) = −d ; in particular, w ∈Ω.
We deduce that
|z −w | =
∫
[zw]
|∇ϕ|dH 1 ≥
∣∣∣∣∫
[zw]
∇ϕ · z −w|z −w | dH
1
∣∣∣∣= ∣∣ϕ(z)−ϕ(w)∣∣= 2d +a ≥ 2d .
890 Radu Ignat
Figure 1. Geometry of characteristics Xx (containing x ′) and X y .
By Pythagoras’s formula in the triangle x ′y w we have |y − w |2 = ℓ2 +d 2. Denoting α the angle
between −−→w y and ∇ϕ(y), the cosine formula in the triangle w y z yields
−cosα= |y −w |
2 +|y − z|2 −|z −w |2
2|y − z| · |y −w |
= 2d
2 +ℓ2 −|z −w |2
2d
p
d 2 +ℓ2
≤− 2d
2 −ℓ2
2d
p
d 2 +ℓ2
=−1+O
(
ℓ2
d 2
)
.
As α ∈ (0, π2 ), it yields sin2α = 1− cos2α ≤ 2(1− cosα) = O( ℓ2d 2 ). So, sinα = O( ℓd ). Denoting
β ∈ (0, π2 ) the angle between −−→w y and ∇ϕ(x), we compute in the triangle x ′y w :
sinβ≤ tanβ= |y −x
′|
|x ′−w | =
ℓ
d
.
In particular, 0 ≤α+β≤ π2 if ℓ< d10 . Finally, denoting γ the angle between ∇ϕ(x) and ∇ϕ(y), the
triangle inequality yields
sinγ≤ sin(α+β) ≤ sinα+ sinβ=O
(
ℓ
d
)
.
We conclude ∣∣∇ϕ(x)−∇ϕ(y)∣∣= 2sin γ
2
=O(sinγ) =O
(
ℓ
d
)
≤ C
d
|x − y |
for some universal C > 0. □
The conclusion of Theorem 1 follows. □
Acknowledgments
The author thanks Xavier Lamy for very useful discussions.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
Radu Ignat 891
References
[1] L. A. Caffarelli and M. G. Crandall, “Distance functions and almost global solutions of
eikonal equations”, Commun. Partial Differ. Equations 35 (2010), no. 3, pp. 391–414.
[2] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton–Jacobi equations, and opti-
mal control, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser,
2004, xiv+304 pages.
[3] C. De Lellis and R. Ignat, “A regularizing property of the 2D-eikonal equation”, Commun.
Partial Differ. Equations 40 (2015), no. 8, pp. 1543–1557.
[4] F. Ghiraldin and X. Lamy, “Optimal Besov differentiability for entropy solutions of the
eikonal equation”, Commun. Pure Appl. Math. 73 (2020), no. 2, pp. 317–349.
[5] R. Ignat, “Two-dimensional unit-length vector fields of vanishing divergence”, J. Funct. Anal.
262 (2012), no. 8, pp. 3465–3494.
[6] P.-E. Jabin, F. Otto and B. Perthame, “Line-energy Ginzburg–Landau models: zero-energy
states”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1 (2002), no. 1, pp. 187–202.
[7] P.-L. Lions, Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathe-
matics, Pitman Advanced Publishing Program, 1982, iv+317 pages.


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Claire Burrin

Comptes Rendus
Mathématique
Claire Burrin
Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem
Volume 363 (2025), p. 1-5
Online since: 6 March 2025
https://doi.org/10.5802/crmath.693
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 1-5
https://doi.org/10.5802/crmath.693
Research article / Article de recherche
Dynamical systems / Systèmes dynamiques
Equidistribution of continuous low-lying pairs
of horocycles via Ratner’s theorem
Équidistribution de paires d’horocycles longs via le
théorème de Ratner
Claire Burrin a
a Institute of Mathematics, University of Zurich, Switzerland
E-mail: claire.burrin@math.uzh.ch
Abstract. We record an alternative proof of a recent joint equidistribution result of Blomer and Michel, based
on Ratner’s topological rigidity theorem. This approach has the advantage of extending to non-uniform
lattices.
Résumé. Nous présentons une preuve alternative d’un résultat récent d’équidistribution jointe de Blomer et
Michel, basée sur le théorème de rigidité topologique de Ratner. Cette approche a l’avantage de s’étendre aux
réseaux non uniformes.
Keywords. Joint equidistribution, measure classification, commensurator, Hecke operators, matrix coeffi-
cients.
Mots-clés. Équidistribution jointe, classification des mesures, commensurateur, opérateurs de Hecke, coef-
ficients matriciels.
2020 Mathematics Subject Classification. 37A10, 37A44, 22F30.
Funding. The author acknowledges the support of the Swiss National Science Foundation (Grant No. 201557).
Manuscript received 1 August 2024, revised 23 September 2024 and 28 September 2024, accepted 28 Septem-
ber 2024.
Expanding closed horocycles are known to become equidistributed on any non-compact
finite-area hyperbolic orbifold, such as, e.g., the low-lying horocycle
{
SL2(Z)
(
x+ iT
)∣∣x ∈ [0,1]} on
the modular curve X = SL2(Z)\H as T →∞. Equidistribution also holds for various arithmetically
interesting growing discrete subsets of points along expanding closed horocycles (see, e.g.,
[2,3,5,9,12]). Most recently Blomer and Michel [1] investigated the joint distribution properties of
sets of the shape {(
SL2(Z)
(
a + i
q
)
,SL2(Z)
(
ab + i
q
))∣∣∣∣a (mod q)}⊂ X ×X (1)
as q → ∞ while b ∈ (Z/qZ)∗ varies with q in a prescribed manner. In parallel, they consider a
“continuous version” of this setup, which establishes the joint equidistribution of two low-lying
horocycles of different speeds under a weak diophantine condition; see Theorem 1 below. By
Dirichlet’s theorem, given any real numbers y and Q > 0, there is a rational number aq so that
|y − aq | < 1qQ and q ≤Q. In the following statement, y is irrational and Q is picked to depend on a
growing parameter T so that as T →∞ we can find a sequence q →∞ for which y has a rational
approximation of that order of magnitude.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
2 Claire Burrin
Theorem 1 (Blomer–Michel). Let X = SL2(Z)\H, T > 1, y ∈ [1,2] and write y = a/q +O(1/(qQ))
for positive coprime integers a, q with q ≤Q := T 0.99. Let I ⊆R be a fixed non-empty interval. Then{(
SL2(Z)
(
x + i
T
)
,SL2(Z)
(
x y + i
T
))∣∣∣∣x ∈ I}⊆ X ×X
equidistributes as T →∞ for pairs (y,T ) with q →∞.1
The argument of proof is based on the estimation of Weyl sums via a shifted convolution
problem, Sato–Tate and a sieving argument, and crucially the measure classification theorem
of Einsiedler–Lindenstrauss [4, Theorem 1.4]. The purpose of this note is to record the following
version of their result, using exclusively homogeneous dynamics. It would be very interesting to
have such an argument for the discrete equidistribution analogue of Theorem 1 concerning sets
of the shape (1), which is the main result of [1] (see [1, Theorem 1.1]).
Write
at =
(p
t 0
0
p
t −1
)
for t ∈R>0 and ux =
(
1 x
0 1
)
for x ∈R.
Theorem 2. Let G = SL2(R), let Γ be a non-uniform lattice in G, set L =G×G,Λ= Γ× Γ. Let T > 1,
and let y > 0 be irrational if Γ is arithmetic or y ̸= 1 if Γ is non-arithmetic, and let I ⊆ R be a fixed
non-empty interval. Then {(
Γux a
−1
T , Γux y a
−1
T
)
: x ∈ I}⊆Λ\L
equidistributes as T →∞.
Proof. Let π be the composition of the smooth immersion ιy : G → L, ιy (g ) = (g , ay g a−1y ) and the
canonical projection L →Λ\L so that for any test function ϕ ∈Cb(Λ\L), we have∫ 1
0
ϕ
(
π(ux a1/T )
)
dx =
∫ 1
0
ϕ
(
Γux a1/T , Γux y a1/T
)
dx,
where we use the standard fact that the expansion (or contraction) of closed horocycles by the
geodesic flow is algebraically expressed by ay ux a−1y = ux y . A result of Shah [13, Theorem 1.4]
asserts that if π(G) =Λ\L then
lim
T→∞
∫ 1
0
ϕ
(
Γux a1/T , Γux y a1/T
)
dx =
∫
Λ\L
ϕ.
We may introduce a conjugation so that L = G ×G , Λ = Γ× a−1y Γay and π corresponds to
the diagonal embedding. Let ∆(G) denote the diagonal embedding of G in L. By Ratner’s
topological rigidity theorem [11], we know that π(G) =π(H) for some closed connected subgroup
H < L containing ∆(G), where H ∩Λ is a lattice in H . Since G is simple, ∆(G) is a maximal
connected subgroup in L and so the only options are that H = L (in which case π(G) =Λ\L and
equidistribution follows) or that H = ∆(G). In the latter case, the requirement that Λ∩ H be a
lattice in H is equivalent to asking that Γy := Γ∩a−1y Γay be a lattice in G . This holds if and only if
ay ∈ Comm(Γ), where Comm(Γ) is the commensurator of Γ in G , i.e.,
Comm(Γ) = {α ∈ GL+2 (R) ∣∣ Γ∩α−1Γα has finite index in both Γ and α−1Γα} .
A non-uniform lattice Γ < G is arithmetic if and only if it is commensurable to SL2(Z). Since
Comm(SL2(Z)) = R× ·GL+2 (Q), we have ay ∈ Comm(Γ) if and only if y ∈Q. If Γ is non-arithmetic,
we claim that ay ∈ Comm(Γ) if and only if y = 1.
By Margulis’ arithmeticity criterium [6, Theorem 1.16, Chapter IX] Comm(Γ) contains Γ as
a finite-index subgroup. In particular, Comm(Γ) is itself a lattice in G and by discreteness, its
maximal unipotent subgroup is infinite cyclic. Let ut be its generator, i.e., ut ′ ̸∈ Comm(Γ) for all
1In the statement of [1, Theorem 1.3], x is replaced by x/T . The argument of proof does however refer to the low-lying
horocycles stated above; see [1, Theorem 4.2 and p. 55].
Claire Burrin 3
|t ′| < t . Let ay ∈ Comm(Γ); up to replacing ay by its inverse, we may assume that y > 1. Then the
fact that a−1y ut ay = ut/y ∈ Comm(Γ) contradicts the minimality of t . □
Remark 3. The statement extends to finitely many factors as follows. If y1, . . . , yn ∈R are chosen
such that ayi /y j ̸∈ Comm(Γ) for i ̸= j , then{(
Γux y1 a
−1
T , . . . , Γux yn a
−1
T
)∣∣x ∈ I}⊂ Γn\Gn
equidistributes as T →∞. This can be compared to [7, Theorem 10.2] and [14, Theorem 4]. See [8]
for a higher rank version, which also relies on commensurability.
In the rest of this note, we come back to the case of X = SL2(Z)\H. By spectral expansion and
Weyl’s equidistribution criterion, equidistribution in Theorem 1 holds if and only if∫ 1
0
ϕ
(
x + i
T
, x y + i
T
)
dx = oϕ(1)
holds for all ϕ = f1 ⊗ f2, where f1 and f2 are non-constant Hecke eigenforms. We will show that
when y ∈Q (and some additional constraints on y) this fails, with a limit that can be expressed as
a product of Hecke eigenvalues and matrix coefficients. To proceed we first recall some notation
connected to the construction of Hecke operators (see, e.g., [10]). For each elementα ∈ Comm(Γ)
we have
ΓαΓ=
M⊔
m=1
Γαhm
where the hm ’s are coset representatives Γ = ⊔Mm=1 Γαhm , with Γα = Γ∩α−1Γα. For every
function f on Γ\G we set ( f | ΓαΓ)(g ) = ∑Mm=1 f (αhm g ). Further, if α ∈ GL+2 (Q) has integer
entries, there is a uniquely determined pair of positive integers l ,m such that l |m and ΓαΓ =
Γ
(
l 0
0 m
)
Γ. The Hecke operator arising from the double coset ΓαΓ is then given by
Tn f = n−1/2
∑
lm=n
l |m
(
f
∣∣∣∣ Γ( l 00 m
)
Γ
)
.
We say that f is a Hecke eigenform if f is an eigenfunction of the Hecke operators Tn , n ≥ 1. We
denote the corresponding Hecke eigenvalues λ f (n).
By abuse of notation, we view functions on X as functions on G that are left SL2(Z)-invariant
and right SO(2)-invariant. We denote by σ1 the usual sum-of-divisors function σ1(n) =∑d |n d .
Proposition 4. Let Γ = SL2(Z). Let y = pq with p, q distinct, coprime and squarefree, and let
ϕ= f1 ⊗ f2 ∈Cb(X ×X ), where f1, f2 are non-constant Hecke eigenforms. Then
lim
T→∞
∫ 1
0
ϕ
(
x + i
T
,
xp
q
+ i
T
)
dx = λ f2 (pq)
p
pq
σ1(pq)
〈
f1, ap/q . f2
〉= λ f1 (pq)ppq
σ1(pq)
〈
aq/p . f1, f2
〉
,
where
〈
f1, ap/q . f2
〉= ∫Γ\G f1(g ) f2(g a−1p/q )dµ(g ) and µ is normalized so that µ(Γ\G) = 1.
Proof. Set Γ = SL2(Z). Since ap/q ∈ Comm(Γ), the group Γp/q = a−1p/q Γap/q ∩ Γ is a finite
index subgroup of Γ; in particular Γp/q < G is a lattice. Then ϕ ◦ ιp/q ∈ Cb(Γp/q \G), and the
equidistribution of pieces of expanding closed horocycles on Γp/q \G yields [14]
lim
T→∞
∫
I
ϕ
(
ux a
−1
T ,uxp/q a
−1
T
)
dx = 1
µ(Γp/q \G)
∫
Γp/q \G
ϕ
(
g , ap/q g a
−1
p/q
)
dµ(g ),
where the Haar measure µ is normalized so that µ(Γ\G) = 1.
For p, q distinct, coprime and squarefree, we find Γap/q Γ= δ−1ppq Γ
(1 0
0 pq
)
Γ, where δx =
( x
x
)
,
and µ(Γp/q \G) = M = σ1(pq) [10, Lemma 4.5.6]. Let F be a fundamental domain for Γ\G ; then
4 Claire Burrin
⊔M
m=1 hmF , where the hm ’s are coset representatives of Γp/q in Γ, is a fundamental domain for
Γp/q \G . We can now compute
1
µ(Γp/q \G)
∫
Γp/q \G
f1(g ) f2(ap/q g a
−1
p/q )dµ(g ) =
1
M
M∑
m=1
∫
hmF
f1(g ) f2(ap/q g a
−1
p/q )dµ(g )
= 1
M
∫
F
f1(g )
M∑
m=1
f2(ap/q hm g a
−1
p/q )dµ(g )
=
p
pq
σ1(pq)
∫
Γ\G
f1(g )Tpq f2(g a
−1
p/q )dµ(g ). □
Acknowledgments
Thanks are due to Alex Gorodnik for organizing the workshop Distribution of orbits: Arithmetics
and Dynamics in the Swiss Alps, where Valentin Blomer’s nice talk motivated this note. Follow-
ing his presentation, Amir Mohammadi and Andreas Strömbergsson both independently also
pointed out the Ratner-based argument presented in this note. The computation that follows
benefitted from separate discussions with Andreas Strömbergsson, and I want to thank Jens
Marklof for references and comments, and the referees for their careful reading and helpful sug-
gestions to improve the exposition of this note.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
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Claire Burrin 5
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Equidistribution of continuous low-lying pairs of horocycles via Ratner’s theorem

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