On lattices with finite Coulombian interaction energy in the plane
OON LATTICES WITH FINITE COULOMBIAN INTERACTIONENERGY IN THE PLANE
YUXIN GE AND ETIENNE SANDIER
Abstract.
We present criteria for the Coulombian interaction energy of infinitely manypoints in the plane with a uniformly charged backgroud introduced in [5] to be finite, aswell as examples. We also show that in this unbounded setting, it is not always possibleto project an L vector field onto the set of gradients in a way that reduces its average L norm on large balls. Introduction
Given a discrete set Λ in the plane (we will also say a lattice ) and a real number m ≥ m ∈ R . It is defined in several steps, following mostly [5].First, denoting ν := (cid:80) p ∈ Λ δ p and for any vector-field j solving(1) − div( j ) = 2 π ( ν − m ) in R , and belonging to L ( R \ Λ , R ) we define W ( j ) as follows: For any R > χ R a smooth approximation of the indicator function of B R , the ball centered at 0 withradius R . More precisely we assume that(2) χ R ≥ (cid:107)∇ χ R (cid:107) ∞ ≤ C , χ R ≡ B R − and χ R ≡ R \ B R , where C is independent of R . Then we let(3) W ( j ) := lim sup R →∞ W ( j, χ R ) | B R | , W ( j, χ R ) := lim sup η → (cid:90) R \∪ p ∈ Λ B ( p,η ) χ R | j | + π log η (cid:88) p ∈ Λ χ R ( p )Second, we consider the set F Λ of vector fields in L ( R \ Λ , R ) satisfying (1) for agiven Λ and m , and the subset P Λ of curl-free vector fields in F Λ , or equivalently the setof those elements in F Λ which are gradients. We may now define(4) W (Λ) := inf ∇ U ∈ P Λ W ( ∇ U ) , ˜ W (Λ) := inf j ∈ F Λ W ( j ) . Note that F Λ and P Λ depend on m , hence so do W (Λ) and ˜ W (Λ). But in fact (see below)the value of m is determined by Λ in the sense that W (Λ) or ˜ W (Λ) can only be finite forat most one value of m (which is the asymptotic density of Λ whenever it exists). In anycase, the value of m will always be clear from the context or made precise. a r X i v : . [ m a t h - ph ] J u l YUXIN GE AND ETIENNE SANDIER
Remark 1.
It will be useful to generalize somewhat the above definition to allow j ’ssatisfying (1) with ν := (cid:88) p ∈ Λ α p δ p . In this case one should modifiy the definition of W ( j, χ R ) : (5) W ( j, χ R ) := lim sup η → (cid:90) R \∪ p ∈ Λ B ( p,η ) χ R | j | + π | α p | log η (cid:88) p ∈ Λ χ R ( p ) . In [5], only W is considered. One could think at first that W and ˜ W are equal, theargument being the following: Since W ( j ) may be seen as the average of | j | over R (withthe infinite part due to the Dirac masses in (1) removed), then projecting onto the set ofcurl-free fields would reduce this quantity, so that the infimum of W ( j ) over F Λ would infact be acheived by some j ∈ P Λ , proving that W (Λ) = ˜ W (Λ). It turns out however thatthis is not the case and in fact we prove (see Theorem 1 below) that with m = 0, Theorem. W ( N ) = + ∞ and ˜ W ( N ) < + ∞ . The rest of the paper is devoted to giving sufficient conditions on Λ for ˜ W and/or W tobe finite. There are roughly two factors which can make W or ˜ W infinite. First, there isthe logarithmic interaction between pairs of points, which can be made infinite by bringingpoints very close to each other: we will not consider this factor here and to rule it out werestrict ourself to uniform Λ’s in the following sense.
Definition 1.
Given a lattice Λ and weights { α p } p ∈ Λ , we say that ν = 2 π (cid:88) p ∈ Λ α p δ p is of uniform type if min p (cid:54) = q ∈ Λ | p − q | > , sup p ∈ Λ | α p | < ∞ . If the weights are all equal to we simply say Λ is of uniform type. The second factor which can make W or ˜ W infinite is the interaction with the back-ground. If we restrict ourselves to uniform Λ’s, then for a given m the quantities W (Λ)or ˜ W (Λ) measure how close (cid:80) p ∈ Λ δ p is to a uniform density m . Our second main resultshows that this can be measured by simply counting the number of points of Λ in anygiven ball (see Theorems 2 and 4). In particular we have Theorem.
Assume that Λ is uniform and that there exists m, C ≥ and ε ∈ (0 , suchthat for any x ∈ R and R > we have, denoting (cid:93)E the number of elements in E , (6) (cid:12)(cid:12) (cid:93) ( B ( x, R ) ∩ Λ) − mπR (cid:12)(cid:12) ≤ CR − ε Then W (Λ) < + ∞ for this value of m . This criterion for finiteness is optimal in the sense that if we replace the right-hand sidein (6) by CR ε , then it is not difficult to construct Λ’s satisfying (6) and having infiniterenormalized energies (see Proposition 5). This criterion can be relaxed a bit in the caseof ˜ W (see Theorem 4). ENORMALIZED ENERGY 3
This leaves open the case ε = 0 (in which case N and Z satisfy (6) with m = 0). In thiscase we are able to prove a partial result (see Theorem 5 for a variant) Theorem.
Let A ⊂ Z and Λ := Z \ A . Assume there exists some constant C > suchthat for all x ∈ R and R > we have (cid:93) ( A ∩ B ( x, R )) ≤ CR.
Then ˜ W (Λ) < + ∞ . The proof of this theorem is based on the fact (see Proposition 6) that under the abovehypothesis there exists a bijection between Z \ A and Z under which points are movedat uniformly bounded distances. This is a discrete analogue of a result of G.Strang [10].The criterion in Theorem 1 is satisfied by perfect (or Bravais) lattices, or more generallyby doubly periodic lattices (see [3]) — even though in this case (see below) the conclusion ofTheorem 1 is almost trivial. However we are not aware that this is known for quasiperiodiclattices, and thus we give a construction similar to that of Theorem 1 which allows us toconclude for an exemple of Penrose-type lattice Λ that ˜ W (Λ) < + ∞ . We have not soughtgenerality in this direction, and refer to Section 6 for the construction of Λ and the proofthat ˜ W (Λ) is finite. 2. Some properties of W , ˜ W We always assume the following property of ν := (cid:80) p ∈ Λ δ p , which is satisfied in particularif Λ is uniform.(7) lim sup R → + ∞ ν ( B R ) | B R | < + ∞ . We begin by recalling some facts from [5, 6].
Structure of P Λ : If Λ satisfies (7) and W (Λ) is finite, then the set {∇ U ∈ P Λ | W ( ∇ U ) < + ∞} is a 2-dimensional affine space. Any two gradients in this setdiffer by a constant vector. Minimization:
For any given m , the function Λ → W defined over the set of Λ’ssatisfying (7) is bounded from below and admits a minimizer. Scaling:
Denote W m the renormalized energy with background m ∈ R . If j satisfies(1) and (7) holds, then W ( j ) = m (cid:16) W ( j (cid:48) ) − π m (cid:17) , with j (cid:48) ( · ) = 1 √ m j (cid:18) ·√ m (cid:19) . Cutoffs:
If (7) holds, then the value of W (Λ) (or ˜ W (Λ)) does not depend on theparticular choice of cut-off functions χ R as long as they satisfy the stated proper-ties. Perfect lattices:
Assume Λ = Z (cid:126)u ⊕ Z (cid:126)v where ( (cid:126)u, (cid:126)v ) is a basis of R satisfying thenormalized volume condition | (cid:126)u ∧ (cid:126)v | = 1. Let Λ ∗ be the dual lattice of Λ. Then,taking m = 1, W (Λ) = π lim x → (cid:88) p ∈ Λ ∗ \{ } e iπp · x π | p | + log | x | − π π. YUXIN GE AND ETIENNE SANDIER
Moreover, the minimum of W among lattices of this type is acheived by the trian-gular lattice Λ := (cid:115) √ (cid:32) (1 , Z ⊕ (cid:32) , √ (cid:33) Z (cid:33) . Uniqueness of m : For a given Λ, there can be at most one value of m for which W (Λ) < + ∞ . Indeed if j (resp. j ) satisfy (1) with m (resp. m ) then − div( j − j ) = m − m , and if m (cid:54) = m this implies that W ( j ) and W ( j ) cannot bothbe finite. To see this one can use Proposition 1 below in case the points in Λ areuniformly spaced. Otherwise one has to resort to the corresponding result in [5].One of the main points in [5, 6] is the fact that W is bounded below. This is in factvery easy to prove in the case of Λ’s — or more generally ν ’s — which are of uniform type.It is a consequence of the following useful fact. Proposition 1. If j satisfies (1) with ν of uniform type, then for any δ < inf p (cid:54) = q ∈ Λ | p − q | there exists g : R → R and C > such that (8) g ≥ − C, such that (9) g = 12 | j | , on R \ ∪ p ∈ Λ B ( p, δ ) , and such that for any compactly supported lipschitz function χ , (10) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R χg − W ( j, χ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CN (cid:107)∇ χ (cid:107) ∞ , where N = (cid:93) { p ∈ Λ | B ( p, δ ) ∩ Supp ∇ χ (cid:54) = ∅ } . Remark 2.
Note that if we take χ such that χ = 1 on B ( p, δ ) and χ = 0 on every other B ( q, δ ) for q (cid:54) = p ∈ Λ then (10) implies that (cid:82) R χg = W ( j, χ ) . This implies in particular,approximating the indicator function B ( p,δ ) by such functions, that for any p ∈ Λ(11) (cid:90) B ( p,δ ) g = W ( j, B ( p,δ ) ) Proof. In R \ ∪ p ∈ Λ B ( p, δ ), we let g = | j | . Then, for any p ∈ Λ and any r ∈ (0 , δ ) suchthat | j | ∈ L ( ∂B ( p, r )) — this is the case for a.e. r — we define λ p,r > λ such that(12) 12 (cid:90) ∂B ( p,r ) min( | j | , λ ) = πα p r − π α p mr. The fact that λ p,r is well defined follows from the fact that the left-hand side of (12) is acontinuous increasing function of λ which increases from 0 to (as λ → + ∞ )12 (cid:90) ∂B ( p,r ) | j | ≥ πr (cid:32)(cid:90) ∂B ( p,r ) j · ν (cid:33) = πr (cid:0) α p − mπr (cid:1) ≥ πα p r − π α p mr. For any r ≤ δ we let, on ∂B ( p, r ), g := 12 ( | j | − λ p,r ) + − πα p m − α p δ log 1 δ . ENORMALIZED ENERGY 5
Then (9) is obviously satisfied, and (8) is satisfied with C = (cid:32) sup p ∈ Λ α p (cid:33) | log δ | δ + π | m | sup p ∈ Λ | α p | . It remains to prove (10). For any function χ and any η ≤ δ we have(13) (cid:90) R \∪ p ∈ Λ B ( p,η ) χ (cid:18) | j | − g (cid:19) = (cid:88) p ∈ Λ (cid:90) B ( p,δ ) \ B ( p,η ) χ (cid:18) | j | − g (cid:19) . Then, writing A for the annulus B ( p, δ ) \ B ( p, η ),(14) (cid:90) A χ (cid:18) | j | − g (cid:19) = χ ( p ) (cid:90) A (cid:18) | j | − g (cid:19) + (cid:90) A ( χ − χ ( p )) (cid:18) | j | − g (cid:19) . We have for any r ≤ δ , on ∂B ( p, r ) (cid:18) | j | −
12 ( | j | − λ p,r ) + (cid:19) = 12 min (cid:0) | j | , λ p,r (cid:1) , hence using (12) we find(15) (cid:90) A (cid:18) | j | − g (cid:19) = (cid:90) δη πα p r − π α p mr dr + π ( δ − η ) (cid:32) πα p m + α p δ log 1 δ (cid:33) = πα p log 1 η − πη α p δ log 1 δ . On the other hand, since | χ − χ ( p ) | ≤ r (cid:107)∇ χ (cid:107) ∞ , and using (12) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) A ( χ − χ ( p )) (cid:18) | j | − g (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107)∇ χ (cid:107) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) δη r (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂B ( p,r ) min (cid:0) | j | , λ p,r (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Cr (cid:33) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107)∇ χ (cid:107) ∞ . This together with (13),(14) and (15) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim η → (cid:90) R \∪ p ∈ Λ B ( p,η ) χ (cid:18) g − | j | (cid:19) − (cid:88) p ∈ Λ πχ ( p ) α p log η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CN (cid:107)∇ χ (cid:107) ∞ , where N = (cid:93) { p ∈ Λ | B ( p, δ ) ∩ Supp ∇ χ (cid:54) = ∅ } . This proves (10). (cid:3) Note that, contrary to the corresponding result in [5], we have not proved that theconstant C in (8) is universal, which is a delicate point. We have included this weakerresult for the sake of self-containedness and because it has a simple proof.3. Examples of finite or infinite energy lattices.
We begin by showing that moving the points in Z at a bounded distance yields a latticeΛ with finite energy, assuming Λ is uniform. Proposition 2.
Let Λ be a lattice in the plane satisfying inf x,y ∈ Λ ,x (cid:54) = y | x − y | > and let Φ : Λ → Z be a bijective map such that sup p ∈ Λ | Φ( p ) − p | < ∞ . Then ˜ W (Λ) < + ∞ , with m = 1 . YUXIN GE AND ETIENNE SANDIER
Proof.
Let R = 2 sup p ∈ Λ | Φ( p ) − p | . Then for every p ∈ Λ, we solve (cid:26) −(cid:52) U p = 2 π (cid:0) δ p − δ Φ( p ) (cid:1) in B ( p, R ) ∂U p ∂ν = 0 on ∂B ( p, R )where ν is the outer unit normal on the boundary. Let V be the Z -periodic solution —which is unique modulo an additive constant — of −(cid:52) V = 2 π (cid:88) p ∈ Z δ p − in R Then by periodicity | V ( x ) + log | x − p || is bounded in C (cid:0) ∪ p ∈ Z B ( p, / (cid:1) , while V ( x ) isbounded in C of the complement. More precisely we have the (see for instance [5]) V ( x ) = (cid:88) p ∈ Z \{ } e iπp · x π | p | Now we define j : R → R by j = ∇ V + (cid:88) p ∈ Λ ∇ U p , where ∇ U p is is extended by 0 outside of B ( p, R ) and thus defined on the whole of R .From the assumptions on Λ and Φ the sum above is finite on any compact set and thus j is well defined and solves − div( j ) = 2 π (cid:88) p ∈ Λ δ p − in R . On the other hand, U p ( x ) + log | x − p | − log | x − Φ( p ) | is bounded in C ( B ( p, R )),uniformly with respect to p ∈ Λ. It follows that j + ∇ log | x − p | is bounded in B ( p, δ )uniformly with respect to p ∈ Λ, and j is bounded in R \ ∪ p ∈ Λ B ( p, δ ), where δ > W ( j ) < + ∞ and then ˜ W (Λ) < + ∞ . (cid:3) We will prove below that the conclusion in the above proposition cannot be improvedto W (Λ) < + ∞ .A consequence of Proposition 2 is Corollary 1.
We have ˜ W ( Z \ Z ) < ∞ , ˜ W ( Z \ N ) < ∞ Proof.
We construct a bijective map from Φ : Z \ Z → Z byΦ( p , p ) = (cid:26) ( p , p −
1) if p ≥ p , p ) if p < Z \ N is similar. (cid:3) A second tool for constructing j ’s with finite energy is ENORMALIZED ENERGY 7
Proposition 3.
Assume j (resp. j ) satisfy (1) with a ν (resp. ν ) of uniform type.Assume also that ν and ν satisfy (7) and that ν + ν is of uniform type.Then, if W ( j ) < ∞ for a background m and W ( j ) < ∞ for the background m , wehave W ( j + j ) < ∞ for the background m + m . First, we prove two lemmas.
Lemma 1.
Assume j satisfies (1) and (7) with ν of uniform type, and assume W ( j ) < ∞ .Then there exists some positive constant C depending on j such that for any R > and δ < inf {| p − q | | p (cid:54) = q ∈ Λ } , (cid:90) B R \∪ p ∈ Λ B ( p,δ ) | j | ≤ CR , (cid:90) B R ∩∪ p ∈ Λ B ( p,δ ) | j − G | ≤ CR , where G ( x ) := α p x − p | x − p | if x ∈ B ( p, δ ) with p ∈ Λ .Proof. Let g be constructed in Proposition 1. From (10), we have (cid:90) χ R g ≤ W ( j, χ R ) + Cn ( R ) ≤ CR , where n ( R ) := (cid:93) (Λ ∩ B R +1 ). Hence(16) (cid:90) χ R g ≤ CR . On the other hand, since g ≥ − C and from the properties of χ R , we have(17) (cid:90) χ R g ≥ (cid:90) B R g − CR ≥ (cid:90) B R \∪ p ∈ Λ B ( p,δ ) | j | + (cid:88) p ∈ Λ ,B ( p,δ ) ⊂ B R (cid:90) B ( p,δ ) g − CR.
For any p ∈ Λ, we define W ( j, B ( p,δ ) ) := lim sup η → (cid:90) B ( p,δ ) \ B ( p,η ) | j | + πα p log η We have, denoting A = B ( p, δ ) \ B ( p, η ),12 (cid:90) A | j | = 12 (cid:90) A | G | + | j − G | + 2 G · ( j − G )= πα p log δη + 12 (cid:90) A | j − G | + α p (cid:90) δη drr (cid:90) ∂B ( p,r ) ν · ( j − G )= πα p log δη + 12 (cid:90) A | j − G | + α p (cid:90) δη drr (cid:90) B ( p,r ) div( j − G )= πα p log δη + 12 (cid:90) A | j − G | + π α p m ( δ − η ) . Hence, we obtain(18) W ( j, B ( p,δ ) ) = lim sup η → (cid:90) A | j | + πα p log η = πα p log δ + 12 (cid:90) B ( p,δ ) | j − G | + π α p mδ Thus, using (11),(19) (cid:90) B ( p,δ ) g = πα p log δ + 12 (cid:90) B ( p,δ ) | j − G | + π α p mδ Gathering (16) to (19), we get CR ≥ (cid:90) χ R g ≥ (cid:90) B R \∪ p ∈ Λ B ( p,δ ) | j | + (cid:88) p ∈ Λ ,B ( p,δ ) ⊂ B R (cid:90) B ( p,δ ) | j − G | − CR . This gives the desired result. (cid:3)
Lemma 2.
Assume j satisfies (1) and (7) with ν of uniform type and let G be the functiondefined in Lemma 1 — for some δ < inf {| p − q | | p (cid:54) = q ∈ Λ } — and extended by on R \ ∪ p ∈ Λ B ( p, δ ) . Then W ( j ) < ∞ ⇔ lim sup R →∞ − (cid:90) B R | j − G | < ∞ where − (cid:82) A denotes the average over A .Proof. The “ = ⇒ ” part of the assertion follows from Lemma 1. We prove the reverseimplication. We denote by g the result of applying Proposition 1 to j .Then from the properties of χ R and using (10), (8), W ( j, χ R ) ≤ (cid:90) B R gχ R + CR ≤ CR + (cid:90) B R \∪ p ∈ Λ B ( p,δ ) g + (cid:88) p ∈ Λ ∩ B R (cid:90) B ( p,δ ) g. Then, as in the proof of Lemma 1, (cid:90) B ( p,δ ) g = W ( j, B ( p,δ ) ) = 12 (cid:90) B ( p,δ ) | j − G | + O (1) . Using this and (9) we find W ( j, χ R ) ≤ CR + 12 (cid:90) B R + δ | j − G | . This yields the desired result. (cid:3)
Proof of Proposition 3.
We denote Λ i the lattice related to j i for i = 1 , j + j . We write ν i = (cid:80) p ∈ Λ i α i,p δp , i = 1 ,
2. Then we choose δ <
12 min (inf {| p − q | | p (cid:54) = q ∈ Λ } , inf {| p − q | | p (cid:54) = q ∈ Λ } , inf {| p − q | | p (cid:54) = q ∈ Λ } ) , and let G i ( x ) = α i,p x − p | x − p | if x ∈ B ( p, δ ) for p ∈ Λ i , and G i = 0 elsewhere.Then, under the assumptions of the proposition, there exists C >
R > (cid:90) B R | j − G | , (cid:90) B R | j − G | < CR . Therefore (cid:90) B R | j + j − ( G + G ) | < CR . In view of the previous Lemma, Proposition 3 is proved. (cid:3)
ENORMALIZED ENERGY 9
Corollary 2.
We have, with m = 0 , ˜ W ( Z ) < + ∞ , ˜ W ( N ) < + ∞ Proof.
There exists j ∈ F Z and from Corollary 1 there exists j ∈ F Z \ Z such that W ( j )and W ( j ) are both finite with m = 1. Then, by Proposition 3 and since − div( j − j ) = (cid:80) p ∈ Z δ p , and Z is uniform, we have W ( j − j ) < + ∞ with m = 0, hence ˜ W ( Z ) < + ∞ .The proof for N is identical. (cid:3) Proposition 4.
For m = 0 we have W ( Z ) < + ∞ , W ( N ) = + ∞ The case of Z . We define V ( x ) := − log | sin( πx ) | . Direct calculations lead to −(cid:52) V = 2 π (cid:88) p ∈ Z δ p in R and |∇ V ( x ) | = π | cos( πx ) || sin( πx ) | . Both V ( x ) and |∇ V ( x ) | are 1-periodic functions. Straightforward calculations yield W ( ∇ V ) < + ∞ . (cid:3) The case of N . We must prove that no ∇ U ∈ P N is such that W ( ∇ U ) < + ∞ . Our strategyis to construct ∇ H ∈ P N such that W ( ∇ H ) = + ∞ , and such that W ( ∇ H , χ R ) 14 ) , (23) (cid:12)(cid:12)(cid:12)(cid:12) ∇ H ( x ) + 1 x − k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (log( | x | + 1) + 1) , in B ( k, 14 ) . For (22), take any x ∈ C \ ∪ k ∈ N B ( k, ), it follows from (21) that |∇ H ( x ) | ≤ (cid:88) ≤ k ≤ [2 | x | +1] (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) k − x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + (cid:88) k> [2 | x | +1] (cid:12)(cid:12)(cid:12)(cid:12) xk ( k − x ) (cid:12)(cid:12)(cid:12)(cid:12) := I + II, where [ · ] denotes the integer part of a real number. We have II ≤ (cid:88) k> [2 | x | +1] | x | ( k − | x | ) ≤ | x | (cid:90) + ∞| x | dtt ≤ , (cid:88) ≤ k ≤ [2 | x | +1] k ≤ (cid:90) | x | +11 dtt ≤ | x | + 1) + 1) . On the other hand, (cid:88) ≤ k ≤ [2 | x | +1] (cid:12)(cid:12)(cid:12)(cid:12) k − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ≤ k ≤ [2 | x | +1] (cid:12)(cid:12)(cid:12)(cid:12) R e ( k − x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) | x | +11 dtt ≤ | x | + 1) + 1) . Therefore, for any x ∈ C \ ∪ k ∈ N B ( k, ), we have |∇ H ( x ) | ≤ | x | + 1) + 1) , andtherefore (22) holds.Now we prove (23). Let x ∈ B ( k, ) for some k ∈ N . As above (cid:12)(cid:12)(cid:12)(cid:12) ∇ H ( x ) + ( R e ( x ) − k, −I m ( x )) | x − k | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x | + 1) + 1) + 1 k ≤ | x | + 1) + 1) , or equivalently, if we use the division of complex number, (cid:12)(cid:12)(cid:12)(cid:12) ∇ H ( x ) + 1 x − k (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x | + 1) + 1) + 1 k ≤ | x | + 1) + 1) , since x ∈ C \ ∪ i (cid:54) = k ∈ N B ( i, ). This proves (23)We now turn to the proof that W ( ∇ H ) = + ∞ . This is done by computing a lowerbound for |∇ H ( x ) | . More precisely we prove that or any ε > 0, there exists some positiveconstant C depending on ε such that(24) |∇ H ( x ) | ≥ (log( | x | + 1) − C ) , if |I m ( x ) | ≥ ε | x | + 1.For this purpose we consider the meromorphic function f ( x ) := (cid:88) k ∈ N xk ( k − x ) . ENORMALIZED ENERGY 11 If |I m ( x ) | ≥ ε | x | + 1, then x ∈ C \ ∪ k ∈ N B ( k, ). Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − (cid:88) ≤ k ≤ [2 | x | +1] (cid:18) k − x − k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ II ≤ , so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + (cid:88) ≤ k ≤ [2 | x | +1] k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ≤ k ≤ [2 | x | +1] (cid:12)(cid:12)(cid:12)(cid:12) k − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ≤ k ≤ [2 | x | +1] (cid:12)(cid:12)(cid:12)(cid:12) I m ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ≤ k ≤ [2 | x | +1] |I m ( x ) |≤ | x | + 1 |I m ( x ) |≤ /ε. On the other hand, we have (cid:88) ≤ k ≤ [2 | x | +1] k ≥ log( | x | + 1) , hence (24) follows. We claim that this implies that W ( ∇ H ) = + ∞ .To see this, we need to bound from below the integral of χ R |∇ H | . We define g byapplying Proposition 1 to ∇ H with δ = 1 / 4. Then we deduce from (8), (9) and the factthat χ R = 1 on B R − that (cid:90) χ R |∇ H | ≥ (cid:90) B R − |∇ H | − CR. Then, integrating (24) on { x ∈ B R − | |I m ( x ) | ≥ ε | x | + 1 } proves that W ( ∇ H ) = + ∞ .We may now argue by contradiction to prove the proposition. Assume that there exists H ∈ P N such that W ( ∇ H ) < + ∞ . Then ¯ H = H − H is a harmonic function over R .For i = 1 , g i by applying Proposition 1 to ∇ H i with δ = 1 / 4. Then CR ≥ W ( ∇ H , χ R ) − W ( ∇ H , χ R ) ≥ (cid:90) χ R ( g − g ) − CR ≥ (cid:90) B R − ( g − g ) − CR. Then, letting G ( x ) = ( x − k ) / | x − k | in B ( k, / 4) for every k and G = 0 outside ∪ k B ( k, / k (cid:90) B ( k, / g i = (cid:90) B ( k, / |∇ H i − G | + C , where C = − π log 4. Together with (9), this implies that (cid:90) B R − ( g − g ) ≥ (cid:90) B R − \∪ k B ( k, / (cid:0) |∇ H | − |∇ H | (cid:1) − [ R ] (cid:88) k =0 (cid:90) B ( k, / |∇ H − G | − CR. Using (23) we have (cid:90) B ( k, / |∇ H − G | ≤ C (log( k + 1) + 1) , so that CR ≥ (cid:90) B R − \∪ k B ( k, / (cid:0) |∇ H | − |∇ H | (cid:1) − CR log R. Then, writing |∇ H | − |∇ H | = |∇ ¯ H | + 2 ∇ ¯ H · ∇ H , we find using (22) that on B R − \ ∪ k B ( k, / |∇ H | − |∇ H | ≥ |∇ ¯ H | − C log R |∇ ¯ H | , and thus, letting A R = B R − \ ∪ k B ( k, / CR ≥ (cid:90) A R (cid:0) |∇ ¯ H | − C log R |∇ ¯ H | (cid:1) − CR log R, from which we easily deduce (cid:90) A R |∇ ¯ H | ≤ CR log R. It follows by a mean value argument that there exists t ∈ [ R/ , R − 1] such that (cid:90) ∂B t |∇ ¯ H | ≤ CR log R, and since ¯ H is harmonic, for any x ∈ B R/ we have |∇ ¯ H ( x ) | ≤ R (cid:90) ∂B t |∇ ¯ H | ≤ C R √ RR log R. Fixing x and letting R → ∞ , we find ∇ ¯ H ( x ) = 0 . Therefore ∇ ¯ H is a constant, which isclearly not possible since W ( ∇ H ) = + ∞ while W ( ∇ H + ∇ ¯ H ) < + ∞ . (cid:3) We summarize the content of this section in the following Theorem 1. We have ˜ W ( Z ) < + ∞ , ˜ W ( N ) < + ∞ , ˜ W ( Z \ Z ) < + ∞ , ˜ W ( Z \ N ) < + ∞ (25) W ( Z ) < + ∞ , W ( Z ) < + ∞ , W ( Z \ Z ) < + ∞ (26) W ( N ) = + ∞ , W ( Z \ N ) = + ∞ (27) Proof. The result comes from Corollary 1, Corollary 2, Proposition 3 and Proposition4. (cid:3) ENORMALIZED ENERGY 13 Sufficient conditions for finite renormalized energy Theorem 2. Given a discrete lattice Λ , assume there exists m ≥ and ε ∈ (0 , , C > such that for any x ∈ R and for R > , we have (28) (cid:12)(cid:12) (cid:93) ( B ( x, R ) ∩ Λ) − mπR (cid:12)(cid:12) ≤ CR − ε and (29) inf x,y ∈ Λ ,x (cid:54) = y | x − y | > Then W (Λ) < + ∞ . Remark 3. For a Bravais lattice, the assumptions in the above theorem are satisfied. Itwas proved by Landau (1915) — see [3] for a more general statement — that the firstassumption holds with ε = 1 / , see [2] for references on more recent developments. We recall a technical lemma. Lemma 3. (Theorem 8.17 in [1] ) Assume q > and p > and v is a solution of thefollowing equation −(cid:52) u = g + (cid:88) i ∂ i f i there exists some constant C such that (cid:107) u (cid:107) L ∞ ( B (0 ,R )) ≤ C ( R − p (cid:107) u (cid:107) L p ( B (0 , R )) + R − q (cid:107) f (cid:107) L q ( B (0 , R )) + R − q (cid:107) g (cid:107) L q/ ( B (0 , R )) ) Proof of Theorem 2. Assume Λ satisfies (28) and (29). The proof consists in constructing j ∈ F Λ such that W ( j ) < + ∞ , which is done by successive approximations constructing afirst some U , then a correction U to U , then a correction U to U + U , etc... In thisconstruction, the U k ’s are functions, and the sum of their gradients will converge to j .Let R n = 2 n − . For all p ∈ Λ, we let U p be the solution to −(cid:52) U p ( y ) = 2 π (cid:18) δ p ( y ) − B ( p,R ) ( y ) πR (cid:19) in B ( p, R ) U p ( y ) = ∂U p ∂ν ( y ) = 0 on ∂B ( p, R )where B ( x,r ) is the indicator function of the ball B ( x, r ). The existence of a solution withNeumann boundary conditions follows from the fact that δ p − B ( p,R πR has zero integral,and the radial symmetry of the solution implies U p is constant on the boundary, and theconstant can be taken equal to zero. In fact, extending U p by zero outside B ( p, R ), weget a solution of −(cid:52) U p ( y ) = 2 π (cid:18) δ p ( y ) − B ( p,R ) ( y ) πR (cid:19) in R , which is supported in B ( p, R ).We let U ( y ) := (cid:88) p ∈ Λ U p ( y ) . This sum is well defined since, Λ being discrete, it is locally finite. Moreover U solves(30) − (cid:52) U ( y ) = 2 π (cid:88) p ∈ Λ δ p − n ( y ) , where n ( y ) := (cid:93) (Λ ∩ B ( y, R )) πR Then we proceed by induction. For any k ≥ U k be the solution to(31) −(cid:52) U kp ( y ) = 2 π (cid:18) B ( p,R k − ) ( y ) πR k − − B ( p,R k ) ( y ) πR k (cid:19) in B ( p, R k ) U kp ( y ) = ∂U kp ∂ν ( y ) = 0 on ∂B ( p, R k ) , and we let U kp = 0 outside the B ( p, R k ). We let U k ( y ) := (cid:80) p ∈ Λ U kp ( y ), so that −(cid:52) U k ( y ) = 2 π ( n k − ( y ) − n k ( y )) , where, for any k ∈ N , n k ( y ) := (cid:93) (Λ ∩ B ( y, R k )) πR k . Now we study the convergence of (cid:80) ∞ k =1 ∇ U k .First we note that there is an explicit formula for U kp . For any k ≥ U kp ( y ) = V (cid:18) y − pR k (cid:19) , where V ( y ) := − | y | + ln 2 if | y | ≤ | y | − ln | y | − if < | y | ≤ 10 if | y | ≥ , from which it follows, since (cid:107)∇ U kp (cid:107) ∞ ≤ CR k and the sum defining U k has at most CR k nonzero terms, that(32) (cid:107)∇ U k (cid:107) ∞ ≤ CR k . Second we estimate (cid:107) U k (cid:107) ∞ . We claim that(33) ∀ k ≥ ∃ C k ∈ R such that (cid:107) U k ( y ) − C k (cid:107) ∞ = O ( R − εk ).Indeed, from (28),(34) (cid:107) n k − m (cid:107) ∞ ≤ CR − − εk . On the other hand, letting n y ( r ) := (cid:93) ( B ( y, r ) ∩ Λ), we have for any y (cid:54)∈ Λ U k ( y ) = (cid:88) p ∈ B ( y,R k ) ∩ Λ V (cid:18) | p − y | R k (cid:19) = (cid:90) R k V (cid:18) tR k (cid:19) n (cid:48) y ( t ) dt = − (cid:90) R k R k V (cid:48) (cid:18) tR k (cid:19) n y ( t ) dt. But, using (28), we have n y ( t ) = mπt + O ( t − ε ), hence U k ( y ) = − mπ (cid:90) R k R k V (cid:48) (cid:18) tR k (cid:19) t dt + O ( R − εk ) . The first term is independent of y , we call it C k . This proves (33).Finally, we note that, from (34), it holds that(35) (cid:107)(cid:52) U k (cid:107) ∞ = O ( R − − εk ) . ENORMALIZED ENERGY 15 Now, we claim that (33) and (35) imply that(36) (cid:107)∇ U k (cid:107) ∞ = O ( R − εk )To see this we use the elliptic estimate of Lemma 3. For all y ∈ R we have(37) (cid:90) B ( y,R k ) |∇ U k | = (cid:90) B ( y,R k ) |∇ ( U k − C k ) | = − (cid:90) B ( y,R k ) (cid:52) U k ( U k − C k )+ (cid:90) ∂B ( y,R k ) ∂U k ∂ν ( U k − C k ) ≤ CR k (cid:16) R − εk + (cid:107)∇ U k (cid:107) ∞ R k − ε (cid:17) Now we apply Lemma 3. For i = 1 , (cid:52) (cid:16) ∂ i U k (cid:17) = − π∂ i ( n k − − n k ) , therefore for any q > p > (cid:107) ∂ i U k (cid:107) L ∞ ( B Rk/ ) ≤ C (cid:16) R k − p (cid:107) ∂ i U k (cid:107) L p ( B Rk ) + R k − q (cid:107) n k − − n k (cid:107) L q ( B Rk ) (cid:17) . Then, taking p = 2 and noting that (34) implies (cid:107) n k − − n k (cid:107) q ≤ CR k q − (1+ ε ) , we findusing (37) that (cid:107) ∂ i U k (cid:107) L ∞ ( B Rk/ ) ≤ C (cid:16) R k − ε + R k − ε (cid:107)∇ U k (cid:107) L ∞ ( B Rk ) (cid:17) + CR k − ε . This proves (36).Now (36) implies that the sum (cid:80) k ≥ ∇ U k converges, and if we let j = ∇ U + (cid:80) k ≥ ∇ U k ,then − div j = 2 π ( (cid:80) p ∈ Λ δ p − m ), using (30), (31) and (34). Moreover j is a gradient sinceit is a sum of gradients, thus j ∈ P Λ .To conclude, it is easy to check, using the assumption inf x,y ∈ Λ ,x (cid:54) = y | x − y | > 0, that W ( ∇ U , χ R ) ≤ CR for all R > 1, and to deduce using (36) that W ( j, χ R ) ≤ CR . (cid:3) For ˜ W the hypothesis of Theorem 2 can be relaxed somewhat. Theorem (cid:48) . Assume there exists some non-negative number m ≥ and some positivenumbers ε ∈ (0 , , C > and a increasing sequence { R n } tending to + ∞ such that forany x ∈ R and for any n ∈ N , we have (cid:12)(cid:12) (cid:93) ( B ( x, R n ) ∩ Λ) − mπR n (cid:12)(cid:12) ≤ CR − εn , and such that (cid:88) n R − εn < + ∞ and inf x,y ∈ Λ ,x (cid:54) = y | x − y | > . Then ˜ W (Λ) < + ∞ . We will use the following simple estimate. Lemma 4. Let u be a solution of the following problem (cid:26) −(cid:52) u = f in Ω ∂u∂ν = 0 on ∂ Ω Then (cid:90) Ω |∇ u | ≤ C | Ω | (cid:107) f (cid:107) ∞ where C is a constant independent of Ω .Proof. We have (cid:90) Ω |∇ u | = − (cid:90) Ω u (cid:52) u = (cid:90) Ω f u ≤ (cid:107) u (cid:107) (cid:107) f (cid:107) ∞ ≤ (cid:112) | Ω |(cid:107) u (cid:107) (cid:107) f (cid:107) ∞ Without loss of generality, we assume (cid:82) u = 0. By Poincar´e inequality, (cid:107) u (cid:107) ≤ C (cid:112) | Ω |(cid:107)∇ u (cid:107) . Finally, the desired result follows. (cid:3) Proof of Theorem (cid:48) . Let µ Λ = (cid:88) p ∈ Λ δ p , I k = B Rk | B R k | , for any integer k > 0, where B Rk is the indicator function of the ball B (0 , R k ).At the first step, for all x ∈ R we let U x be the solution to (cid:40) −(cid:52) U x ( y ) = 2 π ( µ Λ ( y ) − µ Λ ∗ I ( x )) B R ( x − y ) in B ( x, R + 1) ∂U x ∂ν ( y ) = 0 on ∂B ( x, R + 1) . This equation has a solution which is unique up to an additive constant since (cid:90) ( µ Λ ( y ) − µ Λ ∗ I ( x )) B R ( x − y ) dy = µ Λ ∗ ( πR I )( x ) − πR µ Λ ∗ I ( x ) = 0 . We extend ∇ U by zero outside B ( x, R + 1) and let j ( y ) := 1 πR (cid:90) R ∇ U x ( y ) dx, so that − div( j ) = 2 π (cid:88) p ∈ Λ δ p − m ( y ) , where m = µ Λ ∗ I ∗ I . Then we define j k by induction. For x ∈ R we let U kx be the solution to (cid:40) −(cid:52) U kx ( y ) = 2 π ( m k − ( y ) − m k − ∗ I k ( x )) B Rk ( x − y ) in B ( x, R k + 1) ∂U kx ∂ν ( y ) = 0 on ∂B ( x, R k + 1) , and extend ∇ U kx by 0 outside the ball B ( x, R k ). Then we let j k ( y ) := 1 πR k (cid:90) R ∇ U kx ( y ) dx ENORMALIZED ENERGY 17 so that − div( j k )( y ) = 2 π ( m k − ( y ) − m k ( y )) , where m k = m k − ∗ I k ∗ I k . We claim that(38) m k ( y ) = m + O ( R − − εk ) . To see this, it suffices to note that from the commutativity of the convolution we have m k = ( µ Λ ∗ I k ) ∗ ( I k ∗ I k − ∗ I k − ∗ · · · ∗ I ∗ I ) . Then from our first assumption | µ Λ ∗ I k − m | ≤ CR k − (1+ ε ) , which implies (38) since every I k is a positive function with integral 1, and thus convoluting a function with it does notincrease the L ∞ norm.It follows from Lemmas 3 and 4 that for all k ≥ x ∈ R (cid:107)∇ U kx (cid:107) L ∞ ( R ) ≤ CR k − − ε , which yields for all k ≥ (cid:107) j k (cid:107) L ∞ ( R ) ≤ CR k − − ε . Therefore (cid:80) k ≥ (cid:107) j k (cid:107) ∞ ≤ + ∞ and we can define j := (cid:80) k ≥ j k . The vector field j solves − div( j ) = 2 π (cid:88) p ∈ Λ δ p − m in R . Now it suffices to prove that W ( j ) < + ∞ . This is clearly a consequence of the fact that W ( j ) < + ∞ and the fact that (cid:80) k ≥ (cid:107) j k (cid:107) ∞ ≤ + ∞ . On the other hand, W ( j ) < + ∞ isproved as follows: For any p ∈ Λ, and any x ∈ B ( p, R ) we have (cid:107) U x ( y ) − log | y − p |(cid:107) < C in C ( B ( p, δ )) with C, δ > p , x and y , because of the equation satisfiedby U and the uniform spacing of the points in Λ. Also, if x / ∈ B ( p, R ), then (cid:107) U x ( y ) (cid:107) < C in C ( B ( p, δ )).Then, since j = (cid:82) ∇ U x /πR , we have | j ( y ) − log | y − p || < C in B ( p, δ ) for any p ∈ Λand | j | < C outside ∪ p ∈ Λ B ( p, δ ). This implies that W ( j ) < + ∞ . (cid:3) Proposition 5. The conditions in Theorem 2 are optimal in some sense. More precisely,for any m ≥ and any ε > there exists Λ such that ˜ W (Λ) = + ∞ and for any x ∈ R and any R > (cid:12)(cid:12) (cid:93) ( B ( x, R ) ∩ Λ) − mπR (cid:12)(cid:12) ≤ CR ε . Proof. The counter-example is as follows, assuming without loss of generality that ε < / k ∈ N , on the circle ∂B (0 , k ), we distribute uniformly [32 πmk + k ε ] points, where[ x ] is the integer part of x . This is clearly possible maintaining at the same time adistance greater than min(1 / m, 1) (if k is large enough) between the points, since k ε (cid:28) k as k → + ∞ .Then we have as k → + ∞ (cid:93) (Λ ∩ B (0 , k )) − mπ (4 k ) (cid:39) k − (cid:88) i =1 [ i ε ] (cid:39) k ε ε , thus for any j such that − div( j ) = 2 π (cid:88) p ∈ Λ δ p − m , and for any R ∈ (4 k + 1 , k + 3), we have (cid:90) ∂B (0 ,R ) j · ν = 2 π (cid:0) (cid:93) (Λ ∩ B (0 , R )) − mπR (cid:1) (cid:39) π ε k ε . Thus there exist k > c > k > k and for any R ∈ (4 k + 1 , k + 3), wehave(39) 12 (cid:90) ∂B (0 ,R ) | j | ≥ πR (cid:32)(cid:90) ∂B (0 ,R ) j · ν (cid:33) ≥ c k ε Now we construct g using proposition 1 with δ < inf p (cid:54) = q ∈ Λ | p − q | and δ < 1. Forfunctions { χ R } R satisfying (2), we have for any k ∈ N and since the support of χ k +2 doesnot intersect ∪ p ∈ Λ B ( p, δ ) that W ( j, χ k +2 ) = (cid:90) gχ k +2 = (cid:90) ∪ p ∈ Λ B ( p,δ ) gχ k +2 + (cid:90) R \∪ p ∈ Λ B ( p,δ ) gχ k +2 and therefore, since g = | j | outside ∪ p ∈ Λ B ( p, δ ) and g ≥ − C , we obtain W ( j, χ k +2 ) ≥ (cid:88) i ≤ k − (cid:90) B (0 , i +3) \ B (0 , i +1) | j | − CR ≥ CR ε , where we used (39) for the last inequality. Therefore W ( j ) = + ∞ . (cid:3) Critical case In view of Theorem 2 and Proposition 5, the critical discrepancy between (cid:80) p ∈ Λ δ p andthe uniform measure m dx is when (cid:12)(cid:12) (cid:93) ( B ( x, R ) ∩ Λ) − mπR (cid:12)(cid:12) = O ( R ) . This includes thecases Λ = Z or N . As shown by Theorem 1, we cannot expect W (Λ) to be finite undersuch an assumption. However we have the following result for ˜ W . Theorem 3. Let A ⊂ Z and Λ := Z \ A . Assume there exists some constant C > such that for all x ∈ R and for all R > we have (cid:93) ( A ∩ B ( x, R )) ≤ CR. Then ˜ W (Λ) < + ∞ This result is a direct consequence of Proposition 2 and the following: Proposition 6. Let A ⊂ Z . Then the following properties are equivalent. Property I.: There exists some constant C > such that for all x ∈ R and for all R > we have (40) (cid:93) ( A ∩ B ( x, R )) ≤ CR ENORMALIZED ENERGY 19 Property II.: There exists a bijective map Φ : Λ → Z satisfying (41) sup p ∈ Λ | Φ( p ) − p | < ∞ The fact that the second property implies the first one is not difficult. First note that(41) is equivalent to the same property for Φ − , and that Property I is equivalent to thesame property with squares K R of sidelength R replacing the balls of radius R .Now assume (cid:93) ( A ∩ K R ) > CR, then Φ − ( K R ∩ Z ) is included in Z \ A and thuscontains at least CR points which do not belong to K R . Therefore, as C → + ∞ , themaximal distance between an element p of Φ − ( K R ∩ Z ) and K R tends to + ∞ . Thisproves that II = ⇒ I.The proof of the converse is less obvious. It is essentially an application of the max-flow/min-cut duality, with arguments similar in spirit to those found in [10].We let G be a graph for which the set of vertices is Z and the set of edges is A := { ( p, q ) | p, q ∈ Z , (cid:107) p − q (cid:107) = 1 } where (cid:107) · (cid:107) is the Euclidean norm. Given an integer N ∈ N , we define some function µ N : Z → R + p (cid:55)→ N − (cid:93) (Λ ∩ K Np )where K Np := [ kN, ( k + 1) N ) × [ lN, ( l + 1) N ) for p = ( k, l ). Since Λ = Z \ A , µ N is indeednon-negative and µ N ( p ) is equal to (cid:93) ( A ∩ K Np ), i.e. the deficit of the points of Λ in K Np .We introduce the following notions. • A flow , or is a map ϕ : A → R such that for any edge ( p, q ) one has ϕ ( p, q ) = − ϕ ( q, p ). • Given a flow ϕ , its divergence div( ϕ ) is the function div( ϕ ) : Z → R such that forany p ∈ Z one has div( ϕ )( p ) := (cid:88) ( p,q ) ∈A ϕ ( p, q ) • Given a function f : Z → R , its gradient ∇ f is the 1-form ∇ f ( p, q ) = f ( q ) − f ( p ). • Given a subset A of Z , its boundary ∂A is defined by ∂A := { ( p, q ) ∈ A | p ∈ A, q ∈ Z \ A } . • Given a subset A of Z , its perimeter is Per( A ) := (cid:93) ( ∂A ) • A curve connecting p and q is a subset of A of the form { ( p , p ) , ( p , p ) , · · · , ( p n − , p n ) } with p = p and p n = q . A loop or cycle is curve such that p n = p . A graph is connected if any two points can be connected by a curve. • Given a function f : Z → R and B ⊂ Z , its integral on B is defined by (cid:90) B f := (cid:88) p ∈ B f ( p ) . We denote also f ( B ) = (cid:82) B f . • Given a 1-form ϕ and a curve γ , the integral of ϕ on γ is defined by (cid:90) γ ϕ := (cid:88) a ∈ γ ϕ ( a ) • Given two 1-forms ϕ and φ , their inner product is (cid:104) ϕ, φ (cid:105) := 12 (cid:88) a ∈A ϕ ( a ) φ ( a ) • Given a 1-form ϕ and a subset S ⊂ A , the total variation of ϕ with respect to S is defined by [ ϕ, S ] := 12 (cid:88) a ∈ S | ϕ ( a ) | When S = A , we simply write [ ϕ ].We have the following classical results. Lemma 5. (Poincar´e Lemma) Given a 1-form ϕ , if one has (cid:82) γ ϕ = 0 for any loop γ , thenthere exists a function f satisfying ϕ = ∇ f Proof. One fixes some point p ∈ Z and for any q ∈ Z one defines f ( q ) := (cid:82) C ϕ where C isany curve connecting p and q . From the hypothesis, this definition is independent of theparticular curve chosen, and it is easy to check that ϕ = ∇ f . (cid:3) Lemma 6. (Stokes’ formula) Let ϕ be a 1-form with compact support and f be a functionwith compact support. Then one has (cid:104) ϕ, ∇ f (cid:105) = − (cid:90) Z f div ( ϕ ) Proof. We write ϕ as linear combination of elementary 1-forms α ( p,q ) := δ ( { p,q ) } − δ { ( q,p ) } , and note that (cid:104) α ( p,q ) , ∇ f (cid:105) = f ( q ) − f ( p ) = − (cid:90) Z f div( α ( p,q ) ) , since div( α ( p,q ) ) = δ { p } − δ { q } . (cid:3) Lemma 7. (Coarea formula) Let f : Z → R + be a function with the compact support.Then one has [ ∇ f ] = (cid:90) ∞ Per( { f > t } ) dt Proof. We note that [ ∇ f, { ( p, q ) , ( q, p ) } ] = | f ( q ) − f ( p ) | and ∂ { f > t } ∩ { ( p, q ) , ( q, p ) } = ( p, q ) if f ( p ) > t and f ( q ) ≤ t ( q, p ) if f ( q ) > t and f ( p ) ≤ t ∅ otherwise,which implies that (cid:93) ( ∂ { f > t } ∩ { ( p, q ) , ( q, p ) } ) = (cid:26) f ( p ) > t ≥ f ( q ) or f ( q ) > t ≥ f ( p )0 otherwise ENORMALIZED ENERGY 21 Therefore, we get (cid:90) ∞ (cid:93) ( ∂ { f > t } ∩ { ( p, q ) , ( q, p ) } ) dt = | f ( q ) − f ( p ) | . Summing with respect to all couples of edges { ( p, q ) , ( q, p ) } proves the result. (cid:3) We may now set up the duality argument. For any given 1-form ϕ we let (cid:107) ϕ (cid:107) ∞ = sup ( p,q ) ∈A ϕ ( p, q ) = sup {(cid:104) φ, ϕ (cid:105) | φ is compactly supported, [ φ ] ≤ } , and define, α := min − div( ϕ )= µ N max {(cid:104) φ, ϕ (cid:105) | φ ∈ C , [ φ ] ≤ } , where C is the set of compactly supported 1-forms. Lemma 8. One has α = max ∇ f ∈ C , [ ∇ f ] ≤ (cid:90) + ∞ (cid:32)(cid:90) { f>t } µ N − (cid:90) { f< − t } µ N (cid:33) dt Proof. By convex duality, we obtain(42) α = max { φ ∈ C | [ φ ] ≤ } min − div( ϕ )= µ N (cid:104) φ, ϕ (cid:105) . Then given φ ∈ C , we assume there exists a loop γ such that (cid:90) γ φ (cid:54) = 0 . We may then define ϕ t for any t ∈ R by ϕ t ( a ) := (cid:26) t if a ∈ γ γ is a loop, ϕ t has compact support and div( ϕ t ) = 0. Moreover, since (cid:82) γ φ (cid:54) = 0,min t ∈ R (cid:104) φ, ( ϕ + ϕ t ) (cid:105) = −∞ which implies min − div( ϕ )= µ N (cid:104) φ, ϕ (cid:105) = −∞ . As a consequence, the maximum in (42) can be restricted to those φ ’s for which the integralon any loop is zero, i.e. to gradients, in view of Lemma 5. Therefore(43) α = max {∇ f ∈ C | [ ∇ f ] ≤ } min − div( ϕ )= µ N (cid:104)∇ f, ϕ (cid:105) Now, from Lemma 6, we have α = max {∇ f ∈ C | [ ∇ f ] ≤ } min − div( ϕ )= µ N (cid:90) − div( ϕ ) f = max {∇ f ∈ C | [ ∇ f ] ≤ } (cid:90) µ N f On the other hand, for any function f with compact support we have as a well knownconsequence of Fubini’s Theorem (see for instance [4], where this is named the bath-tubprinciple) (cid:90) µ N f + = (cid:90) + ∞ (cid:32)(cid:90) { f>t } µ N (cid:33) dt and (cid:90) µ N f − = (cid:90) + ∞ (cid:16) (cid:90) { f< − t } µ N (cid:17) dt. Together with (43), this proves the result. (cid:3) Lemma 9. Assuming Property I of Proposition 6, there exists C > such that for anyinteger N and any finite B ⊂ Z , we have µ N ( B ) ≤ CN Per( B ) Proof. Let B , · · · , B k be the connected components of B . Then we have disjoint unions B = (cid:83) ki =1 B i and ∂B = (cid:83) ki =1 ∂B i . Set ˜ B i := (cid:83) p ∈ B i K Np . We have µ N ( B i ) = (cid:93) (cid:16) ˜ B i ∩ A (cid:17) , hence µ N ( B i ) ≤ C diam( ˜ B i )Now assume ˜ p = (˜ p , ˜ p ) and ˜ q = (˜ q , ˜ q ) are in ˜ B i and such that diam( ˜ B i ) = (cid:107) ˜ p − ˜ q (cid:107) .Without loss generality, we may assume that (cid:107) ˜ p − ˜ q (cid:107) ≤ p − ˜ q ). There exists p = ( p , p )and q = ( q , q ) in B i such that ˜ p ∈ K Np and ˜ q ∈ K Nq . Moreover,˜ p − ˜ q = N ( p − q ) + ( N − ≤ N ( p − q + 1) . On the other hand, from the connectedness of B i , for any integer x ∈ [ p , q ] we have B i ∩ { r } × Z (cid:54) = ∅ hence writing m x = min { y | ( x, y ) ∈ B i } and M x = max { y | ( x, y ) ∈ B i } ,the two edges (( x, m x ) , ( x, m x − x, M x ) , ( x, M x + 1)) belong to ∂B i . It followsthat Per( B i ) = (cid:93)∂B i ≥ p − q + 1) , and then µ N ( B i ) ≤ CN Per( B i ) , µ N ( B ) = (cid:88) i µ N ( B i ) ≤ CN (cid:88) i Per( B i ) = CN Per( B ) . (cid:3) As a consequence, we obtain Corollary 3. Assuming Property I of Proposition 6, there exists C > and for anyinteger N > there exists a -form ϕ such that (44) − div ( ϕ ) = µ N and for every edge a ∈ A , (45) | ϕ ( a ) | ≤ CN. Proof. It follows from Lemmas 7 and 9 that (cid:90) + ∞ µ N ( { f > t } ) ≤ CN (cid:90) + ∞ Per( { f > t } ) = CN [ ∇ f + ]and (cid:90) + ∞ µ N ( { f < − t } ) ≤ CN (cid:90) + ∞ Per( { f > t } ) = CN [ ∇ f − ] . This implies using Lemma 8 that α ≤ CN max ∇ f ∈ C , [ ∇ f ] ≤ [ ∇ f ] = CN. ENORMALIZED ENERGY 23 Using the definition of α , there exists a 1-form ϕ with the desired properties (changingthe constant to 2 C for instance). (cid:3) Proof of Proposition 6. We construct the bijective map Φ : Λ → Z . This is done byspecifying the for every p, q ∈ Z the number of points in Λ ∩ K Np whose images by Φbelong to Z ∩ K Nq , as follows: n p → q := max( ϕ ( p, q ) , 0) if ( p, q ) ∈ A (cid:93) (Λ ∩ K Np ) − (cid:88) ( p,q ) ∈A n p → q if p = q ϕ is a flow satisfying (44), (45).Now, for the numbers n p → q to indeed correspond to a bijective map Φ we need to checksome of their properties. Property 1. If N is chosen large enough, then for any p, q ∈ Z , we have n p → q ≥ 0. Thisis clear when p (cid:54) = q . In the case p = q , we note that there are exactly 4 edges coming outof p . Thus, from (45) and the fact that (cid:93) (Λ ∩ K Np ) ≥ N − CN we find (with anotherconstant C still independent of N ). n p → p ≥ N − CN. Thus we may indeed choose N large enough so that indeed n p → q ≥ p, q ∈ Z . Property 2. This one is clear from the definition of n p → q : For any p ∈ Z we have (cid:88) q n p → q = (cid:93) (Λ ∩ K Np ) . Property 3. For any q ∈ Z we have (cid:88) p n p → q = N . Indeed, fixing q ∈ Z and all the sums below being with respect to p , (cid:88) p n p → q = n q → q + (cid:88) ( p,q ) ∈A n p → q = (cid:93) (Λ ∩ K Nq ) − (cid:88) ( q,p ) ∈A n q → p + (cid:88) ( p,q ) ∈A n p → q = (cid:93) (Λ ∩ K Nq ) + (cid:88) ( p,q ) ∈A ,ϕ ( p,q ) ≥ ϕ ( p, q ) − (cid:88) ( q,p ) ∈A ,ϕ ( q,p ) ≥ ϕ ( q, p ) . Now since ϕ ( p, q ) = − ϕ ( q, p ) we have (cid:88) ( p,q ) ∈A ,ϕ ( p,q ) ≥ ϕ ( p, q ) − (cid:88) ( q,p ) ∈A ,ϕ ( q,p ) ≥ ϕ ( q, p ) = (cid:88) ( p,q ) ∈A ϕ ( p, q ) = − div ϕ ( q ) . Using (44) this sum is equal to µ N ( q ) = N − (cid:93) (Λ ∩ K Nq ), hence (cid:80) p n p → q = N . The three properties insure that there exists a bijection Φ : Λ → Z such that for any p, q ∈ Z we have n p → q = (cid:93) { x ∈ Λ ∩ K Np | Φ( x ) ∈ Z ∩ K Nq } . Since n p → q (cid:54) = 0 implies (cid:107) p − q (cid:107) ≤ 1, we find that (cid:107) Φ( x ) − x (cid:107) ≤ K N ), for any x ∈ Λ. (cid:3) Remark 4. The conclusion of Theorem 3 holds under the following, less restrictive as-sumption on Λ , which is assumed to be uniform, but not necessarily a subset of Z : i) There exists some positive constant C > such that for any x ∈ R and any R > ,one has | (cid:93) (Λ ∩ B ( x, R )) − πR | ≤ CR. ii) There exists some positive integer N ∈ N such that for any p ∈ Z , one has (cid:93) (cid:0) K N p ∩ Λ (cid:1) ≤ N . Indeed, the second assumption, implies that there exists an injective map Φ p : K N p ∩ Λ → K N p ∩ Z . We define Φ : Λ → Z to be the injective map whose restriction to K N p is Φ p for any p ∈ Z and let Λ = Φ(Λ) . Then Λ is of the form Z \ A , with A satisfying (40) .Theorem 3 implies that ˜ W (Λ ) < + ∞ and then from (2) we deduce that ˜ W (Λ) < + ∞ . We conclude this section with Theorem (cid:48) . Let Λ ⊂ R be discrete and uniform, and of the form Λ = Λ × Z , where Λ ⊂ R .If there exists C > such that for any x ∈ R and R > we have | (cid:93) (Λ ∩ K ( x, R )) − R | ≤ CR — where K ( x, R ) is the square with sidelength R and center x — then ˜ W (Λ) < + ∞ .Proof. The proof of the theorem will follow the same strategy as for Theorem 3, except thatwe work now in one dimension. For any integer N > p ∈ Z we let I Np = [ pN, ( p +1) N )and µ N ( p ) = N − (cid:93) (Λ ∩ I Np ). We consider the graph with Z as the set of vertices and theset of edges A = { ( p, q ) | p, q ∈ Z , | p − q | = 1 } . We claim that there exists C > 0, and for any integer N > ϕ : A → R suchthat(46) − div( ϕ ) = µ N , (cid:107) ϕ (cid:107) ∞ ≤ C. Indeed we define ϕ as follows: ϕ (( k, k + 1)) = k = 0 − k (cid:88) i =1 µ N ( i ) if k ≥ (cid:88) i = k +1 µ N ( i ) if k < . It is clear that − div( ϕ ) = µ N . Moreover, for instance if k ≥ 1, then ϕ (( k, k + 1)) = − k (cid:88) i =1 (cid:0) N − (cid:93) (Λ ∩ I Np ) (cid:1) = (cid:93) (Λ ∩ [ N, ( k + 1) N )) − kN. ENORMALIZED ENERGY 25 But considering the square K = [ N, ( k + 1) N ) × [ N, ( k + 1) N ), we have kN − (cid:93) (Λ ∩ [ N, ( k + 1) N )) = 1 kN (cid:0) ( kN ) − (cid:93) (Λ ∩ K ) (cid:1) , and thus using the hypothesis of the theorem we deduce that | ϕ (( k, k +1)) | ≤ C as claimed.Now we choose N ≥ C + 1 and following the proof of Proposition 6 we can constructa bijective map Φ : Λ → Z such that | Φ( p ) − p | is bounded independently of p . Thisinduces a bijection with the same property from Λ to Z , which proves Theorem 5, usingProposition 2. (cid:3) A Penrose lattice We now describe the construction of a Penrose-type lattice Λ such that ˜ W (Λ) < + ∞ .Of course it would be better to show that Λ satisfies the hypothesis of Theorem 2, butthis to our knowledge an open problem.For the simplicity, we consider the Robinson triangle decompositions in the Penrose’ssecond tilling (P2)–kite and dart tiling, or in the Penrose’s third tilling (P3)–rhombustiling, (for the reference see [8]). The construction is as follows: Ω and Ω are twoRobinson triangles, namely, Ω is an acute Robinson triangle having side lengths (1 , , ϕ ),while Ω is obtuse one with sidelengths ( ϕ, ϕ, ϕ = (1 + √ / 2; the scaled-updomain ϕ Ω decomposes as the union of a copy of Ω and a copy of Ω , where the interiorsare disjoint — and such that ϕ Ω decomposes as the union of one copy of Ω and twocopies of Ω with disjoint interiors (see figure).For i = 1 , p i in the interior of Ω i .Then we proceed by induction, starting with Ω choosing p as the origin, then scalingup by ϕ , then decomposing, then scaling up again, then decomposing each piece, etc...After n steps we have a (large domain) ϕ n Ω decomposed a number of copies of either Ω or Ω . In each copy we have a distinguished point, the union of which is denoted Λ n . As n → + ∞ and modulo a subsequence, Λ n converges to a discrete set Λ, which is uniformsince the distance between two point is no less than min ( d ( p , ∂ Ω ) , d ( p , ∂ Ω )) . Theorem 4. We have ˜ W (Λ) < + ∞ .Proof. For each n we define a current j n as follows. On each copy of Ω i we let j n be equalto (a copy of) ∇ U i , where (cid:40) −(cid:52) U i = δ p i − | Ω i | in Ω i∂U i ∂ν = 0 on ∂ Ω i . Then j n converges as n → + ∞ to a current j such that the following holds in R − div( j ) = (cid:88) p ∈ Λ δ p − α, where α = 1 / | Ω i | on each copy of Ω i . It is not difficult to check that W ( j ) < + ∞ , butthe background density α is not constant. We need to add a correction to j , which is theobject of the following Lemma 10. There exist m ∈ R and a solution of the following equation in R (47) − div ( j (cid:48) ) = α − m ϕ ϕϕ Ω Ω Ω Ω Ω Ω Ω ϕ Ω ϕ Ω Figure 1. such that (cid:107) j (cid:48) (cid:107) ∞ < + ∞ , Assuming the lemma is true we let ˜ j = j + j (cid:48) . Then − div(˜ j ) = (cid:80) p ∈ Λ δ p − m thus˜ j ∈ F Λ for the background m , and the fact that W ( j ) < + ∞ and j (cid:48) ∈ L ∞ implies that W (˜ j ) < + ∞ and the Theorem. (cid:3) Proof of Lemma 10. The current j (cid:48) is obtained as the limit of j n , where j n solves(48) (cid:26) − div( j n ) = α n − m n in ϕ n Ω j n · ν = 0 on ∂ ( ϕ n Ω ) , where α n : ϕ n Ω → R is the function equal to 1 / | Ω i | on each of the copies of Ω i , i = 1 , ϕ n Ω , and where m n is equal to the average of α n on ϕ n Ω .The current j n is defined recursively. First we define the equivalent of α n for Ω -typedomains: For any integer n we tile ϕ n Ω by one copy of ϕ n − Ω and two copies of ϕ n − Ω ,then we tile each of the three pieces, etc... until we have tiled ϕ n Ω by copies of either Ω or Ω . then we let β n : ϕ n Ω → R be the function equal to 1 / | Ω i | on each of the copies ofΩ i , i = 1 , 2. We also define q n to be the equivalent of m n , i.e. the average of β n on ϕ n Ω . ENORMALIZED ENERGY 27 Finally we define ¯ n to be the equivalent of j n for type 2 domains, i.e. the solution of (48)with α n replaced by β n , m n replaced by q n and Ω replaced by Ω .Below it will be convenient to abuse notation by writing ϕ n Ω i for a copy of ϕ n Ω i . Thenwe have ϕ n Ω = ϕ n − Ω ∪ ϕ n − Ω . We let(49) j n = j n − ϕ n − Ω + ¯ n − ϕ n − Ω + ∇ U n ϕ n Ω , where(50) (cid:26) −(cid:52) U n = ( m n − m n − ) ϕ n − Ω + ( m n − q n − ) ϕ n − Ω in ϕ n Ω ∂U n ∂ν = 0 on ∂ ( ϕ n Ω ) . It is straightforward to check that j n satisfies (48) assuming j n − and ¯ n − do.The relation (49) is the recursion relation which repeated n times allows to write j n asequal to a sum of on the one hand error terms ∇ U k (or their type 2 equivalent that wedenote V k ), for k between 1 and n , and on the other hand of a vector field which on eachelementary tile of type Ω of ϕ n Ω is equal to j and on a tile of type Ω is equal to ¯ .However from (48) we may take j = 0 and ¯ = 0, thus we are left with evaluating theerror terms. Claim: There exists C > k > (cid:107)∇ U k (cid:107) ∞ , (cid:107)∇ V k (cid:107) ∞ ≤ Cϕ − k . This clearly proves that the sum of errors for k = 1 . . . n is bounded in L ∞ independentlyof n and therefore that { j n } is bounded in L ∞ . Then the limit j (cid:48) is in L ∞ .To prove the lemma, it remains to prove the claim, and to show that j (cid:48) satisfies (47)for some m ∈ R , which in view of (48) amounts to showing that { m n } n converges. Forthis we define u n (resp. u n +1 ) be the number of elementary tiles of type Ω (resp. Ω )in ϕ n Ω . We define similarly v n and v n +1 by replacing Ω by Ω . Therefore u = 1, u = 0, v = 0, v = 1. We have the following recurrence relations u n +2 = u n + u n +1 , u n +3 = u n + 2 u n +1 , which we can summarize as the single relation u n +2 = u n +1 + u n . Similarly v n +2 = v n +1 + v n . It follows that u n = ϕ n ϕ + 2 + ( − ϕ ) − n ϕ + 1 ϕ + 2 , v n = ϕ n ϕϕ + 2 + ( − ϕ ) − n − ϕϕ + 2 . We have u n = aϕ n + O ( ϕ − n ) and v n = bϕ n + O ( ϕ − n ) with a = ϕ +2 and b = ϕϕ +2 strictlypositive. Then we easily deduce that m n = u n + u n +1 u n | Ω | + u n +1 | Ω | = m + O ( ϕ − n ) , where m = 1 + ϕ | Ω | + ϕ | Ω | , and similarly that q n = m + O ( ϕ − n ) . This proves in particular the convergence of { m n } n .Moreover it shows that the right-hand side of (50) is bounded by Cϕ − n . By ellipticregularity (lemma 3 and lemma 4) we deduce that (cid:107)∇ U n (cid:107) ∞ ≤ C | ϕ n Ω | ϕ − n = C | Ω | ϕ − n , and a similar bound for V n . This proves the claim, and the lemma (cid:3) Remark 5. The above construction could easily be generalized to similar recursive con-structions. Acknowledgments. The authors wish to thank Y.Meyer for helpful discussions. References [1] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order , Reprint of the1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.[2] F.G¨otze, Lattice point problems and values of quadratic forms , Invent. math. 157, 195-226 (2004).[3] E. Hlawka, ber Integrale auf konvexen K¨orpern I, II , Monatsh. Math. 54, 136, 81-99 (1950).[4] E. Lieb and M. Loss, Analysis , Graduate Studies in Mathematics, 14. American Mathematical Society,Providence, RI, 1997. xviii+278 pp.[5] E. Sandier and S. Serfaty, From the Ginzburg-Landau model to vortex lattice problems , Comm. Math.Phys. 313 (2012), 635-743[6] E. Sandier and S. Serfaty, 2D Coulomb Gases and the Renormalized Energy , arXiv:1201.3503.[7] E. Sandier and S. Serfaty, 1D Log Gases and the Renormalized Energy: Crystallization at VanishingTemperature , arXiv:1303.2968.[8] M. Senechal, Quasicrystals and geometry , Cambridge University Press, Cambridge, 1995.[9] S. Serfaty and I. Tice, Lorentz space estimates for the Coulombian renormalized energy , Commun.Contemp. Math. 14, 1250027 (2012)[10] G. Strang, Maximum flows and minimum cuts in the plane . J. Global Optim. 47 (2010), no. 3, 527-535. Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, CNRS UMR 8050, D´epartementde Math´ematiques, Universit´e Paris Est-Cr´eteil Val de Marne, 61 avenue du G´en´eral deGaulle, 94010 Cr´eteil Cedex, France E-mail address : [email protected] Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, CNRS UMR 8050, D´epartementde Math´ematiques, Universit´e Paris Est-Cr´eteil Val de Marne, 61 avenue du G´en´eral deGaulle, 94010 Cr´eteil Cedex, France E-mail address ::