Shifted lattices and asymptotically optimal ellipses
SSHIFTED LATTICES AND ASYMPTOTICALLY OPTIMALELLIPSES
RICHARD S. LAUGESEN AND SHIYA LIU
Abstract.
Translate the positive-integer lattice points in the first quadrant bysome amount in the horizontal and vertical directions. Take a decreasing concave(or convex) curve in the first quadrant and construct a family of curves by rescalingin the coordinate directions while preserving area. Consider the curve in the familythat encloses the greatest number of the shifted lattice points: we seek to identifythe limiting shape of this maximizing curve as the area is scaled up towards infinity.The limiting shape is shown to depend explicitly on the lattice shift. The resultholds for all positive shifts, and for negative shifts satisfying a certain condition.When the shift becomes too negative, the optimal curve no longer converges to alimiting shape, and instead we show it degenerates.Our results handle the p -circle x p + y p = 1 when p > < p < p -circle generates the family of p -ellipses, andso in particular we identify the asymptotically optimal p -ellipses associated withshifted integer lattices.The circular case p = 2 with shift − / p = 1) generates an open problem about minimizing high eigenvalues of quantumharmonic oscillators with normalized parabolic potentials. Introduction
Among all ellipses centered at the origin with given area, consider the one enclosingthe maximum number of positive integer lattice points. Does it approach a circularshape as the area tends to infinity? Antunes and Freitas [2] showed the answer isyes. We tackle a variant of the problem in which the lattice is translated by someincrements in the x - and y -directions, and show the asymptotically optimal ellipseis no longer a circle but an ellipse whose semi-axis ratio depends explicitly on thetranslation increments. This optimal ratio succeeds in “balancing” the horizontaland vertical empty strip areas created by the translation of the lattice; see Figure 1.The precise statement is given in Theorem 2.Generalized ellipses obtained by stretching a (concave or convex) smooth curvecan be handled by our methods too, in Theorem 5. The results hold for all positivetranslations, and for small negative translations that satisfy a computable, curve-dependent criterion. Date : October 5, 2018.2010
Mathematics Subject Classification.
Primary 35P15. Secondary 11P21, 52C05.
Key words and phrases.
Translated lattice, concave curve, convex curve, p -ellipse, spectral opti-mization, Dirichlet Laplacian, Schr¨odinger eigenvalues, harmonic oscillator. a r X i v : . [ m a t h . SP ] J u l SHIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES
Figure 1.
An ellipse that maximizes (among ellipses with the samearea) the number of enclosed positive-integer lattice points shifted by4 units horizontally and 2 units vertically. This optimal ellipse roughlybalances the areas of the horizontal and vertical empty strips; see The-orem 2.When the curve is a straight line, one arrives at an open problem for right trianglesthat contain the most lattice points. The shape of these triangles exhibits a surpris-ing clustering behavior as the area tends to infinity, as revealed by our numericalinvestigations in Section 9. This clustering conjecture has been investigated recentlyin the unshifted case by Marshall and Steinerberger [21].Section 10 motivates this paper by connecting to spectral minimization problemsfor the Dirichlet Laplacian, and raises conjectures for the quantum harmonic oscillatorand for a whole family of such Schr¨odinger eigenvalue problems. The recent advanceson high eigenvalue minimization began with work of Antunes and Freitas [1, 2, 3, 10],and continued with contributions from van den Berg, Bucur and Gittins [6], van denBerg and Gittins [7], Bucur and Freitas [8], Gittins and Larson [11], Larson [17], andMarshall [20].We show in Section 10 that the original result of Antunes and Freitas does notextend to the subclass of symmetric eigenvalues. Instead, the optimal rectangle de-generates in the limit.
Remark.
The lattice point counting estimates in this paper are similar to those usedfor the Gauss circle problem, which aims for accurate asymptotics on the countingfunction inside the circle (and other closed curves) as the area grows to infinity. Thebest known error estimate on the Circle Problem is due to Huxley [14].The lattice counting formulas in our paper differ somewhat from that work, be-cause we consider only one quadrant of lattice points and our regions contain emptystrips due to the translation of the lattice. Further, we focus on proving suitable in-equalities (rather than asymptotics) for the counting function, in order to prevent themaximizing shape from degenerating. In essence, we develop inequalities that tradeoff the empty regions in the vertical and horizontal directions. After degenerationhas been ruled out, we can invoke asymptotic formulas with error terms that neednot be as good as Huxley’s in order to prove convergence to a limiting shape.
HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 3 Results
Consider a strictly decreasing curve Γ lying in the first quadrant with x - and y -intercepts at L . Represent the curve as the graph of y = f ( x ) where f is strictlydecreasing for x ∈ [0 , L ]. Denote the inverse function of f as g ( y ) for y ∈ [0 , L ]. Nowcompress the curve by a factor s > x -direction and stretch it by the samefactor in the y -direction: Γ( s ) = graph of sf ( sx ) . Next scale the curve Γ( s ) by a factor r > r Γ( s ) = graph of rsf ( sx/r ) . Given numbers σ, τ > −
1, consider the translated or shifted positive-integer lattice( N + σ ) × ( N + τ ) , which lies in the open first quadrant. Define the shifted-lattice counting functionunder the curve s Γ( s ) to be N ( r, s ) = number of shifted positive-integer lattice points lying inside or on r Γ( s )= (cid:8) ( j, k ) ∈ N × N : k + τ ≤ rsf (cid:0) ( j + σ ) s/r (cid:1)(cid:9) . The set S ( r ) consists of s -values that maximize N ( r, s ), that is, S ( r ) = argmax s> N ( r, s ) , r > . Write x − = (cid:40) , x ≥ , | x | , x < . Our first theorem will say that the maximizing set S ( r ) is bounded, under eitherof the following conditions on the shift parameters σ, τ > − Parameter Assumption 2.1.
Γ is concave and strictly decreasing, withmax (cid:110) f (cid:0) − σ − − σ − L (cid:1) , g (cid:0) − τ − − τ − L (cid:1)(cid:111) < (cid:16) − σ − − τ − (cid:17) L. (1) Parameter Assumption 2.2.
Γ is convex and strictly decreasing, withmin (cid:110) (1 − σ − ) f (cid:0) − σ − − σ − L (cid:1) , (1 − τ − ) g (cid:0) − τ − − τ − L (cid:1)(cid:111) > σ − + τ − ) L (2)and µ f ( σ ) def = min (cid:110) (1 + σ ) f (cid:0) σ σ x (cid:1) − f ( x ) : 1 + σ σ L ≤ x ≤ L (cid:111) > , (3) µ g ( τ ) def = min (cid:110) (1 + τ ) g (cid:0) τ τ y (cid:1) − g ( y ) : 1 + τ τ L ≤ y ≤ L (cid:111) > . (4) SHIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES
When σ, τ ≥
0, conditions (1) and (2) hold automatically (using that 0 < f ( x ) < L and 0 < g ( y ) < L when x, y ∈ (0 , L )) and conditions (3) and (4) also hold (using that f and g are strictly decreasing and positive). Thus the Parameter Assumptions aresignificant only when σ < τ <
0. They are used to obtain an upper bound onthe counting function: see the comments after Proposition 10 and Proposition 15.
Theorem 1 (Uniform bound on optimal stretch factors) . If the curve Γ and the shiftparameters σ, τ > − / satisfy Parameter Assumption 2.1 or 2.2, then for each ε > one has S ( r ) ⊂ (cid:2) B ( τ, σ ) − − ε, B ( σ, τ ) + ε (cid:3) for all large r ,where B ( σ, τ ) = 2 + σ + τ + (cid:112) (2 + σ + τ ) − σ + 1 / τ )2( σ + 1 / . The bounding constant B ( σ, τ ) depends only on the shift parameters, not on thecurve Γ. The bounding constant B (0 ,
0) = 4 in the unshifted case is consistent withour earlier work [18, Theorem 2].Theorem 1 is proved in Section 6. Note it does not assume the curve is smooth.If the curve is smooth, then the optimal stretch set S ( r ) for maximizing the latticecount is not only bounded but converges asymptotically to a computable value, asstated in the next theorem. First we state the smoothness conditions to be used. Concave Condition 2.3.
Γ is concave, and for some ( α, β ) ∈ Γ with α, β > f ∈ C [0 , α ] , g ∈ C [0 , β ], with f (cid:48) < , α ], f (cid:48)(cid:48) < , α ], f (cid:48)(cid:48) monotonic on [0 , α ], g (cid:48) < , β ], g (cid:48)(cid:48) < , β ], g (cid:48)(cid:48) monotonic on [0 , β ]. Convex Condition 2.4.
Γ is convex, and for some ( α, β ) ∈ Γ with α, β > f ∈ C [ α, L ] , g ∈ C [ β, L ], with f (cid:48) < α, L ), f (cid:48)(cid:48) > α, L ], f (cid:48)(cid:48) monotonic on [ α, L ], g (cid:48) < β, L ), g (cid:48)(cid:48) > β, L ], g (cid:48)(cid:48) monotonic on [ β, L ]. Theorem 2 (Sufficient conditions for asymptotic balance of optimal curve) . If thecurve Γ and shift parameters σ, τ > − / satisfy either Parameter Assumption 2.1and Concave Condition 2.3, or Parameter Assumption 2.2 and Convex Condition 2.4,then the stretch factors maximizing N ( r, s ) approach s ∗ = (cid:115) τ + 1 / σ + 1 / as r → ∞ , with S ( r ) ⊂ [ s ∗ − O ( r − / ) , s ∗ + O ( r − / )] , and the maximal lattice count has asymptotic formula max s> N ( r, s ) = r Area(Γ) − rL (cid:112) ( σ + 1 / τ + 1 /
2) + O ( r / ) . HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 5
Figure 2.
The family of p -circles x p + y p = 1, for 0 < p < ∞ . Exam-ple 3 and Example 4 consider p = 2 and p = 1 /
2, respectively.
In particular, when the shift parameters σ and τ are equal, the optimal stretch factorsfor maximizing N ( r, s ) approach s ∗ = 1 as r → ∞ . The theorem follows from Theorem 5 below, which has weaker hypotheses.We call the optimally stretched curve ( s = s ∗ ) “asymptotically balanced” in termsof the shift parameters, because the optimal shape balances the areas of the emptystrips that are created by translation of the lattice: a horizontal rectangle of width rL/s ∗ and height τ + 1 / rs ∗ L and width σ + 1 /
2. (The “+1 /
2” arises from thinking of each lattice point as thecenter of a unit square.) Further, subtracting these two areas, each of which equals rL (cid:112) ( σ + 1 / τ + 1 / r correction term inthe theorem. Example 3 (Sufficient condition on shift parameters for the circle) . When the curveΓ is the portion of the unit circle in the first quadrant, one takes L = 1 , f ( x ) = √ − x , and α = β = 1 / √
2. Notice f is smooth and concave, with monotonicsecond derivative. By symmetry it suffices to consider σ ≤ τ . When σ ≤ τ ≤ (cid:114) − (cid:0) σ σ (cid:1) < σ + 2 τ + 1 . When σ ≤ ≤ τ , equality in Parameter Assumption 2.1 would give a straight line.The resulting allowable region of ( σ, τ )-shift parameters for Theorem 2 is plotted onthe left side of Figure 3. Example 4 (Sufficient condition on shift parameters for p -circle with p = 1 / . Suppose Γ is the part of the 1 / L =1 , f ( x ) = (1 − x / ) , and take α = β = 1 /
4. Notice f is smooth and convex, withmonotonic second derivative f (cid:48)(cid:48) ( x ) = x − / . The region of allowable shift parametersfor Theorem 2 can be found numerically from Parameter Assumption 2.2, as shownon the right side of Figure 3. SHIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES
Figure 3.
The allowable shift parameters ( σ, τ ) for Theorem 2 formthe regions above the plotted curves, in the special cases where Γ isa circle (figure on the left) and a p -circle with p = 1 / − .
06 (left figure) and − .
04 (right figure). The straight lines in the second and fourth quad-rants are vertical and horizontal, respectively. The curves joining theintercepts are not quite straight lines. See Example 3 and Example 4.Next we define weaker smoothness conditions. Let ( α, β ) be a point on the curveΓ with α, β > Weaker Concave Condition 2.5.
Suppose Γ is concave, and: • f ∈ C (0 , α ] , f (cid:48) < , f (cid:48)(cid:48) <
0, and a partition 0 = α < α < · · · < α l = α exists such that f (cid:48)(cid:48) is monotonic on ( α i − , α i ) for each i = 1 , , . . . , l ; • g ∈ C (0 , β ] , g (cid:48) < , g (cid:48)(cid:48) <
0, and a partition 0 = β < β < · · · < β m = β exists such that g (cid:48)(cid:48) is monotonic on ( β i − , β i ) for each i = 1 , , . . . , m ; • positive functions δ ( r ) and (cid:15) ( r ) exist such that δ ( r ) = O ( r − a ) , f (cid:48)(cid:48) (cid:0) δ ( r ) (cid:1) − = O ( r − a ) , (5) (cid:15) ( r ) = O ( r − b ) , g (cid:48)(cid:48) (cid:0) (cid:15) ( r ) (cid:1) − = O ( r − b ) , (6)as r → ∞ , for some numbers a , a , b , b > • and define a = 1 / , b = 1 / f (cid:48)(cid:48) ( x ) cannot be too small as x → Weaker Convex Condition 2.6.
Suppose Γ is convex, and: • f ∈ C [ α, L ) , f (cid:48) < , f (cid:48)(cid:48) >
0, and a partition α = α < α < · · · < α l = L exists such that f (cid:48)(cid:48) is monotonic on ( α i − , α i ) for each i = 1 , , . . . , l ; • g ∈ C [ β, L ) , g (cid:48) < , g (cid:48)(cid:48) >
0, and a partition β = β < β < · · · < β m = L exists such that g (cid:48)(cid:48) is monotonic on ( β i − , β i ) for each i = 1 , , . . . , m ; • positive functions δ ( r ) and (cid:15) ( r ) exist such that δ ( r ) = O ( r − a ) , f (cid:48)(cid:48) (cid:0) L − δ ( r ) (cid:1) − = O ( r − a ) , (7) (cid:15) ( r ) = O ( r − b ) , g (cid:48)(cid:48) (cid:0) L − (cid:15) ( r ) (cid:1) − = O ( r − b ) , (8) HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 7 as r → ∞ , for some numbers a , a , b , b > • and suppose f ( x ) = L + O ( x a ) as x →
0, and g ( y ) = L + O ( y b ) as y → a , b > δ ( r ) = (cid:15) ( r ) = r − , a = b = 1 / a = b = 1 /
4, and noting that f (cid:48)(cid:48) (0) (cid:54) = 0 , g (cid:48)(cid:48) (0) (cid:54) = 0.The same reasoning shows Convex Condition 2.4 implies Weaker Convex Condi-tion 2.6 with a = b = 1 /
4, since g ( L ) = 0 , g (cid:48) ( L ) ≤ , g (cid:48)(cid:48) ( L ) > ⇒ g ( L − y ) ≥ cy for small y > c >
0, and substituting y = (cid:112) x/c gives L − f ( x ) ≤ (cid:112) x/c for small x >
0, andsimilarly for g .Thus Theorem 2 follows immediately from the next result. Theorem 5 (Weaker conditions for asymptotic balance of optimal curve) . If the curve Γ and shift parameters σ, τ > − / satisfy either Parameter Assumption 2.1 andWeaker Concave Condition 2.5, or Parameter Assumption 2.2 and Weaker ConvexCondition 2.6, then the stretch factors maximizing N ( r, s ) approach s ∗ = (cid:115) τ + 1 / σ + 1 / as r → ∞ , with S ( r ) ⊂ (cid:2) s ∗ − O ( r −E ) , s ∗ + O ( r −E ) (cid:3) where E = min { , a , a , a , b , b , b } . Further, the maximal lattice count has asymptotic formula max s> N ( r, s ) = r Area(Γ) − rL (cid:112) ( σ + 1 / τ + 1 /
2) + O ( r − E ) . (9)The proof in Section 7 relies on lattice point counting propositions developed inSection 3, Section 4 and Section 5. Example 6 ( p -circles) . Suppose Γ is the part of the p -circle | x | p + | y | p = 1 lying inthe first quadrant. When p > < p < a = b = p/ f ( x ) =1 + O ( x p ) as x → g ( y ) = 1 + O ( y p ) as y → p -circle, p (cid:54) = 1. The allowable shift parameters canbe determined numerically from Parameter Assumption 2.1 or 2.2, as in Example 3and Example 4.Next we show there can be no “universal” allowable region of negative shifts forTheorem 2. Specifically, for each choice of negative shifts σ, τ <
0, no matter howclose to zero, a curve exists whose optimal stretch parameters grow to infinity orshrink to 0 as r → ∞ . That is, the optimal curve degenerates in the limit. SHIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES
Theorem 7 (Negative shift: optimal concave curve can degenerate) . If − < σ < , τ > − , then a concave C -smooth curve Γ exists, with intercepts at L = 1 , suchthat for each (cid:15) ∈ (0 , one has S ( r ) ⊂ (0 , r (cid:15) − ) ∪ ( r − (cid:15) , ∞ ) (10) for all large r . The construction is given in Section 8. The point of the theorem is that as soonas one of the shift parameters is negative, a concave curve exists for which the maxi-mizing stretch parameters approach either 0 or ∞ as r → ∞ .For convex curves, we do not know an analogue of Theorem 7: does a universalallowable region of ( σ, τ ) parameters exist in which Theorem 2 holds for all C -smoothconvex decreasing curves?The “bad” curve in Theorem 7 can even be a quarter circle: Proposition 8 (Negative shift: the optimal ellipse can degenerate) . If the curve Γ is the quarter unit circle, and σ, τ > − with either σ ≤ − / or τ ≤ − / , then foreach (cid:15) ∈ (0 , one has S ( r ) ⊂ (0 , r (cid:15) − ) ∪ ( r − (cid:15) , ∞ ) for all large r . The proof is in Section 8. And in Section 10 we apply this result to Laplacianeigenvalue minimization on rectangles.3.
Concave curves — counting function estimates
In order to prove Theorem 5 we need to estimate the counting function. The curveΓ is taken to be concave decreasing in the first quadrant, throughout this section.Denote the horizontal and vertical intercepts by x = L and y = M respectively,where L and M are positive but not necessarily equal. Allowing unequal interceptsis helpful for some of the results below.We start with a preliminary r -dependent bound on the maximizing set S ( r ). Theproof of this bound also makes clear why N ( r, s ) attains its maximum as a functionof s , for each fixed r , so that the set S ( r ) is well defined. Lemma 9 (Linear-in- r bound on optimal stretch factors for concave curves) . If σ, τ > − then S ( r ) ⊂ (cid:2) (1 + τ ) /rM, rL/ (1 + σ ) (cid:3) whenever r ≥ (2 + σ + τ ) / √ LM .Proof. The curve r Γ( s ) with the particular choice s = (cid:112) L/M has horizontal and ver-tical intercepts equal to r √ LM . That intercept value is ≥ (2 + σ + τ ), by assumptionon r in this lemma. Hence by concavity, r Γ( s ) encloses the point (1 + σ, τ ) and so N ( r, s ) > s , which means the maximum of s (cid:55)→ N ( r, s )is greater than 0.When s > rL/ (1 + σ ), the x -intercept of r Γ( s ) is less than 1 + σ and so no shiftedlattice points are enclosed, meaning N ( r, s ) = 0. Thus the maximum is not attainedfor such s -values. Arguing similarly with the y -intercept shows the maximum is alsonot attained when s < (1 + τ ) /rM . The lemma follows. (cid:3) HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 9
Figure 4.
Concave curve enclosing lattice points shifted in the nega-tive direction. The square areas represent the lattice point count, whilethe triangles estimate the discrepancy between that count and the areaunder the curve, as needed for Proposition 10.The last lemma required only that Γ be concave decreasing. Smoothness was notneeded. Smoothness is not used in the next proposition either, which gives an upperbound on the counting function and so extends a result from the unshifted case [18,Proposition 10].
Proposition 10 (Two-term upper bound on counting function for concave curves) . Let σ, τ > − . The number N ( r, s ) of shifted lattice points lying inside r Γ( s ) satisfies N ( r, s ) ≤ r Area(Γ) − C rs + σ − τ − (11) for all r ≥ (1 − σ − ) s/L and s ≥ , where C = C (Γ , σ, τ ) = 12 (cid:16) M − f ( 1 − σ − − σ − L ) (cid:17) − σ − M − τ − L. (12)The constant C might or might not be positive. Parameter Assumption 2.1 consistsof the assumption C > g , in thesituation where L = M . Proof.
First suppose σ ≤ , τ ≤
0. Write N for the number of shifted lattice pointsunder Γ, and suppose L ≥ σ so that (cid:98) L − σ (cid:99) ≥
1. Extend the curve Γ horizontallyfrom (0 , M ) to ( σ, M ), so that f ( σ ) = M . Construct triangles with vertices at (cid:0) i − σ, f ( i − σ ) (cid:1) , (cid:0) i + σ, f ( i + σ ) (cid:1) , (cid:0) i − σ, f ( i + σ ) (cid:1) for i = 1 , . . . , (cid:98) L − σ (cid:99) ,as illustrated in Figure 4. The rightmost vertex of the final triangle has horizontalcoordinate (cid:98) L − σ (cid:99) + σ , which is less than or equal to L . These triangles lie abovethe unit squares with upper right vertices at shifted lattice points, and lie below thecurve Γ due to concavity. Hence N + Area(triangles) ≤ Area(Γ) − σ ( M − τ ) − τ ( L − σ ) − στ, (13)where the correction terms on the right side of the inequality represent the areas ofthe rectangular regions outside the first quadrant. Letting k = (cid:98) L − σ (cid:99) ≥
1, we computeArea(triangles) = k (cid:88) i =1 (cid:0) f ( i − σ ) − f ( i + σ ) (cid:1) = 12 (cid:0) M − f ( k + σ ) (cid:1) ≥ (cid:0) M − f ( 1 + σ σ L ) (cid:1) (14)because f is decreasing and k + σ ≤ L < k + 1 + σ implies k + σ > k + σk + 1 + σ L ≥ σ σ L. Combining (13) and (14) proves N ≤ Area(Γ) − σM − τ L − (cid:16) M − f ( 1 + σ σ L ) (cid:17) + στ. (15)Now we replace Γ with the curve r Γ( s ), meaning we replace N, L, M, f ( x ) with N ( r, s ) , rs − L, rsM, rsf ( sx/r ) respectively, thereby obtaining the desired estimate(11) (noting that L/s ≤ Ls since s ≥ L ≥ σ becomes r ≥ (1 + σ ) s/L under the rescaling, and so we have proved the proposition in the case σ ≤ , τ ≤ σ > , τ >
0, the number of shifted lattice points inside r Γ( s ) is less than orequal to the number when there is no shift ( σ = τ = 0), simply because the curve isdecreasing. Thus this case of the proposition follows from the “ σ, τ ≤
0” case above.When σ > , τ ≤
0, the number of shifted lattice points inside r Γ( s ) is less thanor equal to the number for σ = 0 with the same τ value, and so this case of theproposition follows also from the “ σ, τ ≤
0” case above. A similar argument holdswhen σ ≤ , τ > (cid:3) Corollary 11 (Improved two-term upper bound on counting function for concavecurves) . Let σ, τ > − . If s is bounded above and bounded below away from , as r → ∞ , then the number N ( r, s ) of shifted lattice points lying inside r Γ( s ) satisfies N ( r, s ) ≤ r Area(Γ) − r (cid:0) s − τ L + s ( σ + 1 / M (cid:1) + o ( r ) . (16) Proof.
Take c > c − < s < c throughout the rest of the proof.Suppose σ, τ ≤
0. Let K ≥
1. Repeat the proof of Proposition 10 except with theinitial supposition L ≥ σ replaced by L ≥ K + σ , and do not assume s ≥
1. Onefinds N ( r, s ) ≤ r Area(Γ) − D K rs − τ Lrs − + στ for all r ≥ ( K + σ ) s/L , where D K = D K (Γ , σ ) = 12 (cid:16) M − f ( K + σK + 1 + σ L ) (cid:17) + σM. HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 11
We deducelim sup r →∞ sup s 0. That proves the corollary when σ, τ ≤ σ > , τ ≤ 0. We will relate this case to the previous one. To emphasizethe dependence of the counting function on the shift parameters, write N σ,τ ( r, s ) forthe counting function that was previously written N ( r, s ). Adding columns of shiftedlattice points at x = σ − (cid:100) σ (cid:101) + 1 , . . . , σ − , σ gives the counting function N (cid:101) σ,τ ( r, s )where (cid:101) σ = σ − (cid:100) σ (cid:101) ∈ ( − , N (cid:101) σ,τ ( r, s ) = N σ,τ ( r, s ) + (cid:100) σ (cid:101)− (cid:88) i =0 (cid:98) rsf (cid:0) s ( σ − i ) /r (cid:1) − τ (cid:99) , = N σ,τ ( r, s ) + (cid:100) σ (cid:101) rsM + o ( r ) , as r → ∞ , since s is bounded above and f is continuous with f (0) = M . Since (cid:101) σ, τ ≤ 0, we may apply (16) with σ replaced by (cid:101) σ to obtain N (cid:101) σ,τ ( r, s ) ≤ r Area(Γ) − r (cid:0) s − τ L + s ( σ − (cid:100) σ (cid:101) + 1 / M (cid:1) + o ( r ) as r → ∞ . Combining the above two formulas, we prove the corollary for σ > , τ ≤ σ ≤ , τ > 0, simply add the appropriate rows instead of columns and arguelike above using (cid:100) τ (cid:101) instead of (cid:100) σ (cid:101) , and using the boundedness of s − . Similarly, onecan treat the case σ > , τ > (cid:3) The next proposition gives an asymptotic approximation to N ( r, s ), assuming thecurve is concave decreasing and has suitably monotonic second derivative. Proposition 12 (Two-term counting estimate for concave curves) . Let σ, τ > − and ≤ q < . If Weaker Concave Condition 2.5 holds and s + s − = O ( r q ) then N ( r, s ) = r Area(Γ) − r (cid:0) s − ( τ + 1 / L + s ( σ + 1 / M (cid:1) + O ( r Q ) (17) as r → ∞ , where Q = max { , + q, − a + q, − a + q, − b + q, − b + q } . Special cases: (i) If q = 0 then Q = 1 − e where e = min { , a , a , b , b } .(ii) If Concave Condition 2.3 holds then Q = max { , + q } . The numbers a , a , b , b come from Weaker Concave Condition 2.5. That Con-dition also involves a point ( α, β ) ∈ Γ with α, β > 0, which we use in the followingproof. Figure 5. Curve r Γ( s ) enclosing positive-integer lattice points shiftedby ( σ, τ ) = ( − . , − . (cid:101) O , and (cid:101) L and (cid:102) M are thenew x - and y -intercepts, as defined in the proof of Proposition 12. Proof. The idea is to translate and truncate the curve r Γ( s ) as in Figure 5, in orderto reduce to an unshifted lattice problem. Then we invoke known results from ourearlier paper [18] (which builds on work of Kr¨atzel [15, 16] and a theorem of van derCorput).Step 1 — Translating and truncating. Notice rs → ∞ and rs − → ∞ as r → ∞ ,since s = O ( r q ) and s − = O ( r q ) with q < 1. Thus by taking r large enough, weinsure rs − g (cid:16) s − τr (cid:17) > rs − α > σ, rsf (cid:16) s σr (cid:17) > rsβ > τ. For all large r one also has δ ( r ) < α and (cid:15) ( r ) < β , by Weaker Concave Condition 2.5.Given a large r satisfying the above conditions, and a corresponding s > 0, we let (cid:101) α = rs − α − (1 + σ ) , (cid:101) β = rsβ − (1 + τ ) , and (cid:101) L = rs − g (cid:16) s − τr (cid:17) − (1 + σ ) , (cid:102) M = rsf (cid:16) s σr (cid:17) − (1 + τ ) , so that 0 < (cid:101) α < (cid:101) L, < (cid:101) β < (cid:102) M . Consider the point (cid:101) O = (1 + σ, τ ) in the first quadrant. Regard this point asthe new origin, and let (cid:101) Γ be the portion of r Γ( s ) lying in the new first quadrant —see Figure 5. That is, (cid:101) Γ is the graph of (cid:101) f ( x ) = rsf (cid:16) s x + 1 + σr (cid:17) − (1 + τ ) , ≤ x ≤ (cid:101) L, and also of its inverse function (cid:101) g ( y ) = rs − g (cid:16) s − y + 1 + τr (cid:17) − (1 + σ ) , ≤ y ≤ (cid:102) M . HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 13 Notice ( (cid:101) α, (cid:101) β ) ∈ (cid:101) Γ, since f ( (cid:101) α ) = (cid:101) β . Write (cid:101) N for the number of positive-integer latticepoints under the curve (cid:101) Γ. That is, (cid:101) N = { ( j, k ) ∈ N × N : k ≤ (cid:101) f ( j ) } . This (cid:101) N does not count the lattice points in the first column or row, which arise from j = 0 or k = 0.Weaker Concave Condition 2.5 guarantees that (cid:101) f is C -smooth on the interval[0 , (cid:101) α ], with (cid:101) f (cid:48) < (cid:101) f (cid:48)(cid:48) < (cid:101) g is C -smooth on [0 , (cid:101) β ] with (cid:101) g (cid:48) < (cid:101) g (cid:48)(cid:48) < , (cid:101) α ] as 0 = (cid:101) α < (cid:101) α < · · · < (cid:101) α (cid:101) l = (cid:101) α where theinterior partition points are chosen to be the elements of { rs − α i − (1 + σ ) : i = 1 , . . . , l − } that happen to lie between 0 and (cid:101) α . Observe (cid:101) f (cid:48)(cid:48) is monotonic on each subinterval ofthe partition, by Weaker Concave Condition 2.5. Similarly, (cid:101) g (cid:48)(cid:48) is monotonic on eachsubinterval of the corresponding partition 0 = (cid:101) β < (cid:101) β < · · · < (cid:101) β (cid:101) m = (cid:101) β of the interval[0 , (cid:101) β ].Let (cid:101) δ = (cid:2) rs − δ ( r ) − (1 + σ ) (cid:3) + , (cid:101) (cid:15) = (cid:2) rs(cid:15) ( r ) − (1 + τ ) (cid:3) + , so that 0 ≤ (cid:101) δ < (cid:101) α and 0 ≤ (cid:101) (cid:15) < (cid:101) β .To relate some of these old and new quantities, we denote antiderivatives of f, g by F ( x ) = (cid:90) x f ( t ) d t, G ( y ) = (cid:90) y g ( t ) d t, (18)and observe thatArea( (cid:101) Γ) = r Area(Γ) − r (cid:0) F ((1 + σ ) s/r ) + G ((1 + τ ) s − /r ) (cid:1) + (1 + σ )(1 + τ ) , (cid:101) f (cid:48) ( x ) = s f (cid:48) (cid:0) s x + 1 + σr (cid:1) , (cid:101) f (cid:48)(cid:48) ( x ) = s r f (cid:48)(cid:48) (cid:0) s x + 1 + σr (cid:1) , (cid:90) (cid:101) α | (cid:101) f (cid:48)(cid:48) ( x ) | / d x = r / (cid:90) α (1+ σ ) s/r | f (cid:48)(cid:48) ( x ) | / d x ≤ r / (cid:90) α | f (cid:48)(cid:48) ( x ) | / d x, (cid:101) l (cid:88) i =1 | (cid:101) f (cid:48)(cid:48) ( (cid:101) α i ) | / ≤ l (cid:88) i =1 r / s − / | f (cid:48)(cid:48) ( α i ) | / , and similarly for (cid:101) g except with s replaced by s − . Step 2 — Estimating the counting function. Applying part (a) of [18, Proposi-tion 12] to the curve (cid:101) Γ and using the preceding relationships, we get (cid:12)(cid:12) (cid:101) N − r Area(Γ) + r (cid:0) F ((1 + σ ) s/r ) + G ((1 + τ ) s − /r ) (cid:1) + r (cid:0) sf ((1 + σ ) s/r ) + s − g ((1 + τ ) s − /r ) (cid:1)(cid:12)(cid:12) ≤ r / (cid:16) (cid:90) α | f (cid:48)(cid:48) ( x ) | / d x + (cid:90) β | g (cid:48)(cid:48) ( y ) | / d y (cid:17) + 175 r / (cid:0) s − / | f (cid:48)(cid:48) ( δ ( r )) | / + s / | g (cid:48)(cid:48) ( (cid:15) ( r )) | / (cid:1) + 525 r / (cid:0) l (cid:88) i =1 s − / | f (cid:48)(cid:48) ( α i ) | / + m (cid:88) j =1 s / | g (cid:48)(cid:48) ( β j ) | / (cid:1) + 14 ( l (cid:88) i =1 s | f (cid:48) ( α i ) | + m (cid:88) j =1 s − | g (cid:48) ( β j ) | )+ r s − δ ( r ) + s(cid:15) ( r )) + l + m + 12 (1 + σ ) + 12 (1 + τ ) + (1 + σ )(1 + τ ) + 1 , (19)where we dealt with the term involving | (cid:101) f (cid:48)(cid:48) ( (cid:101) δ ) | − / in [18, Proposition 12] as follows.One has (cid:101) f (cid:48)(cid:48) ( (cid:101) δ ) = r − s f (cid:48)(cid:48) ( z ) where z = r − s ( (cid:101) δ + 1 + σ ) ≥ δ ( r ), and so by mono-tonicity of f (cid:48)(cid:48) on each subinterval of the partition (as assumed in Weaker ConcaveCondition 2.5) one concludes | (cid:101) f (cid:48)(cid:48) ( (cid:101) δ ) | ≥ r − s min {| f (cid:48)(cid:48) (cid:0) δ ( r ) (cid:1) | , | f (cid:48)(cid:48) (cid:0) α (cid:1) | , . . . , | f (cid:48)(cid:48) (cid:0) α l (cid:1) |} . Thus the term involving | (cid:101) f (cid:48)(cid:48) ( (cid:101) δ ) | − / can be estimated by the sum of terms involving | f (cid:48)(cid:48) (cid:0) δ ( r ) (cid:1) | − / and | f (cid:48)(cid:48) (cid:0) α i (cid:1) | − / .The right side of (19) already has the desired order O ( r Q ), by direct estimationand using that s + s − = O ( r q ) and 2 q < + q since q < N ( r, s ) and (cid:101) N count the same lattice points,except that N ( r, s ) also counts the points in the first row and column. That is, (cid:101) N = N ( r, s ) − (cid:98) rsf (cid:0) (1 + σ ) s/r (cid:1) − τ (cid:99) − (cid:98) rs − g (cid:0) (1 + τ ) s − /r (cid:1) − σ (cid:99) + 1= N ( r, s ) − rsf (cid:0) (1 + σ ) s/r (cid:1) − τ − rs − g (cid:0) (1 + τ ) s − /r (cid:1) − σ + ρ ( r, s )for some number ρ ( r, s ) ∈ [1 , rsf ((1 + σ ) s/r ) = rsM + O ( s ) ,r F ((1 + σ ) s/r ) = rs (1 + σ ) M + O ( s ) , and similarly for g and G . The proposition now follows straightforwardly, since O ( s ) = O ( r q ). (cid:3) Lemma 13. If f is decreasing and concave on [0 , L ] then f ( x ) = f (0) + O ( x ) , F ( x ) = f (0) x + O ( x ) , as x → ,where F ( x ) = (cid:82) x f ( t ) d t is the antiderivative of f ( x ) . HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 15 Proof. The difference quotient ( f ( x ) − f (0)) /x is a decreasing function of x since f isconcave, and it is less than or equal to 0 since f is decreasing. Hence the differencequotient is bounded, and so f ( x ) = f (0) + O ( x ). Integrating completes the proof. (cid:3) Convex curves — counting function estimates Assume the curve Γ is convex decreasing, throughout this section. We will proveestimates for convex curves analogous to the work in Section 3 for concave curves.Lemma 14 below is an improved r -dependent bound on the optimal stretch fac-tors, generalizing Ariturk and Laugesen’s lemma from the unshifted situation [5,Lemma 7.2]. By “improved” we refer to the upper and lower bounds: for instance,when σ = 0 the upper bound in Lemma 14 improves on the bound in Lemma 9 by afactor of 2. This tighter bound on the optimal stretch factor gives us more flexibilitywhen deriving the two-term counting estimate in Proposition 15.In the next lemma we assume for simplicity that the x - and y -intercepts are both L , so that we need not change the definitions of µ f ( σ ) and µ g ( τ ) in Section 2. Lemma 14 (Improved linear-in- r bound on optimal stretch factors for convex curves) . If σ, τ > − with µ f ( σ ) > and µ g ( τ ) > , then S ( r ) ⊂ (cid:104) τrL , rL σ (cid:105) whenever r ≥ max (cid:16) (2 + σ ) (cid:113) τ ) /Lµ f ( σ ) , (2 + τ ) (cid:113) σ ) /Lµ g ( τ ) (cid:17) . (20) Proof. Claim 1: N ( r, s ) = 0 if s ∈ (cid:0) , (1+ τ ) /rL (cid:3) or s ∈ (cid:2) rL/ (1+ σ ) , ∞ (cid:1) . Indeed, the curve r Γ( s ) has x - and y -intercepts at rL/s and rsL , respectively, and so if rL/s ≤ σ or rsL ≤ τ then the point (1 + σ, τ ) is not enclosed by the curve and so thelattice count N ( r, s ) is zero.Claim 2: if (20) holds and s ∈ (cid:0) rL/ (2 + σ ) , rL/ (1 + σ ) (cid:1) then N ( r, s ) < N (cid:16) r, σ σ s (cid:17) . To prove this claim, notice the x -intercept satisfies1 + σ < rLs < σ, and so only the first column of shifted lattice points (the points with x -coordinate at1+ σ ) can contribute to the count inside r Γ( s ). Hence N ( r, s ) = (cid:98) rsf ((1+ σ ) s/r ) − τ (cid:99) .Meanwhile, if we count shifted lattice points in the first two columns (where x = 1 + σ and x = 2 + σ ) we find N (cid:16) r, σ σ s (cid:17) (21) ≥ (cid:4) rs σ σ f (cid:16) (1 + σ ) s (2 + σ ) r (cid:17) − τ (cid:5) + (cid:4) rs σ σ f (cid:16) (1 + σ ) sr (cid:17) − τ (cid:5) > rs σ σ f (cid:16) (1 + σ ) s (2 + σ ) r (cid:17) + rs σ σ f (cid:16) (1 + σ ) sr (cid:17) − τ − rsf (cid:16) (1 + σ ) sr (cid:17) + rs σ (cid:16) (1 + σ ) f (cid:16) (1 + σ ) s (2 + σ ) r (cid:17) − f (cid:16) (1 + σ ) sr (cid:17)(cid:17) − τ ) ≥ rsf (cid:16) (1 + σ ) sr (cid:17) + rs σ µ f ( σ ) − τ ) > rsf (cid:16) (1 + σ ) sr (cid:17) ≥ N ( r, s ) , where to get the final line we use that rs σ µ f ( σ ) > τ ), which follows from s > rL/ (2 + σ ) and the lower bound on r in (20). The proof of Claim 2 is complete.Claim 3: if (20) holds and s ∈ (cid:0) (1 + τ ) /rL, (2 + τ ) /rL (cid:1) then N ( r, s ) < N (cid:16) r, τ τ s (cid:17) . The proof is analogous to Claim 2, except counting in rows instead of columns.Claim 4: if (20) holds then the maximizing s -values for N ( r, s ) lie in the interval (cid:2) (2 + τ ) /rL, rL/ (2 + σ ) (cid:3) . To see this, note that N ( r, s (cid:48) ) > s (cid:48) > 0, bythe strict inequality in Claim 2, and so the maximum does not occur in the intervalsconsidered in Claim 1. The maximum does not occur in the interval considered inClaim 2, as that claim itself shows, and similarly for Claim 3. Thus the maximummust occur in the remaining interval, which proves Claim 4 and thus finishes theproof of the lemma. (cid:3) The next bound generalizes work of Ariturk and Laugesen [5, Proposition 5.1] fromthe unshifted situation ( σ = τ = 0) to the shifted case. Proposition 15 (Two-term upper bound on counting function for convex curves) . Let σ, τ > − . The number N ( r, s ) of shifted lattice points lying inside r Γ( s ) satisfies N ( r, s ) ≤ r Area(Γ) − C rs + σ − τ − (22) for all r ≥ (2 − σ − ) s/L and s ≥ , where C = C (Γ , σ, τ ) = 12 (1 − σ − ) f ( 1 − σ − − σ − L ) − σ − M − τ − L. The constant C need not be positive. That is why hypothesis (2) in ParameterAssumption 2.2 includes (for L = M ) the assertion that C > Proof. First consider σ ≤ , τ ≤ 0. Write N for the number of shifted lattice pointsunder Γ. Suppose L ≥ σ . Extend the curve horizontally from (0 , M ) to ( σ, M ), so HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 17 Figure 6. Convex curve enclosing lattice points shifted in the negativedirection. The square areas represent the lattice point count, while thetriangles and trapezoid estimate the discrepancy between that countand the area under the curve in Proposition 15that f ( σ ) = M . Construct a trapezoid (see Figure 6) with vertices at (cid:0) σ, f (1 + σ ) (cid:1) , (cid:0) σ, f (1 + σ ) (cid:1) , (0 , h ), ( σ, h ) where h = f (1 + σ ) − (1 + σ ) f (cid:48) (1 + σ ). Also constructtriangles with vertices (cid:0) i − σ, f ( i + σ ) (cid:1) , (cid:0) i + σ, f ( i + σ ) (cid:1) , (cid:0) i − σ, f ( i + σ ) − f (cid:48) ( i + σ ) (cid:1) ,where i = 2 , . . . , (cid:98) L − σ (cid:99) . These triangles lie above the squares with upper rightvertices at the shifted lattice points, and like below the curve by convexity, as Figure 6illustrates. Hence N + Area(trapezoid and triangles) ≤ Area(Γ) − σ ( M − τ ) − τ ( L − σ ) − στ (23)Let k = (cid:98) L − σ (cid:99) ≥ 2, so that k + σ ≤ L < k + σ + 1. ThenArea(trapezoid) = 12 (base + top) · (height)= − 12 (1 − σ ) · (1 + σ ) f (cid:48) (1 + σ ) ≥ 12 (1 + σ ) (cid:0) f (1 + σ ) − f (2 + σ ) (cid:1) by convexity, and using that 1 − σ ≥ 1. Further, convexity impliesArea(triangles) = − k (cid:88) i =2 f (cid:48) ( i + σ ) ≥ k − (cid:88) i =2 (cid:0) f ( i + σ ) − f ( i + 1 + σ ) (cid:1) + 12 (cid:0) f ( k + σ ) − f ( L ) (cid:1) = 12 f (2 + σ ) . (24) Hence Area(trapezoid) + Area(triangles) (25) ≥ 12 (1 + σ ) f (1 + σ ) − σf (2 + σ ) ≥ 12 (1 + σ ) f ( 1 + σ σ L ) − σf ( 2 + σ σ L ) (26)since f is decreasing and L/ (2 + σ ) ≥ 1. Combining (23) and (26) and using f ( L ) = 0proves N ≤ Area(Γ) − σM − τ L − 12 (1 + σ ) f (cid:0) σ σ L (cid:1) + στ. Now replace Γ with the curve r Γ( s ), meaning replace N, L, M, f ( x ) with N ( r, s ), rs − L , rsM , rsf ( sx/r ) respectively. Using s ≥ 1, we know L/s ≤ Ls ; the assumption L ≥ σ becomes r ≥ (2 + σ ) s/L . Thus we obtain (22) in the case σ ≤ , τ ≤ (cid:3) Corollary 16 (Improved two-term upper bound on counting function for convexcurves) . Let σ, τ > − . If s is bounded above and bounded below away from , as r → ∞ , then the number N ( r, s ) of shifted lattice points lying inside r Γ( s ) satisfies N ( r, s ) ≤ r Area(Γ) − r (cid:0) s − τ L + s ( σ + 1 / M (cid:1) + o ( r ) . (27) Proof. Fix c > c − < s < c in the rest of the proof.Suppose σ, τ ≤ 0, and let K ≥ 2. Repeat the proof of Proposition 15 except withthe initial requirement L ≥ σ replaced by L ≥ K + σ , and do not assume s ≥ N ( r, s ) ≤ r Area(Γ) − E K rs − τ Lrs − + στ. for all r ≥ ( K + σ ) s/L , where E K = E K (Γ , σ ) = 12 (1 + σ ) f (cid:0) σK + σ L (cid:1) − σf ( 2 + σK + σ L ) + σM. Hence lim sup r →∞ sup s 0, which proves the corollary when σ, τ ≤ σ and τ . (cid:3) In the next proposition we state a two-term asymptotic for lattice point countingunder convex curves. HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 19 Proposition 17 (Two-term counting estimate for convex curves) . Let σ, τ > − . IfWeaker Convex Condition 2.6 holds and s + s − = O (1) then N ( r, s ) = r Area(Γ) − r (cid:0) s − ( τ + 1 / L + s ( σ + 1 / M (cid:1) + O ( r − E ) (28) as r → ∞ , where E = min { , a , a , a , b , b , b } . In particular, if Convex Condi-tion 2.4 holds then (28) holds with E = . Proposition 17 does not assume the intercepts L and M are equal, and so wemodify Weaker Convex Condition 2.6 by taking each occurrence of “ L ” that relatesto the function g and changing it to “ M ”, and changing the a -condition to f ( x ) = M + O ( x a ). Proof. We use the idea from Proposition 12: translate and truncate the curve r Γ( s )to reduce to an unshifted lattice problem, and then use results from Ariturk andLaugesen’s paper [5].Assume r Γ( s ) does not pass through any point in the shifted lattice. This assump-tion will be removed in the final step of the proof.Step 1 — Translating and truncating. Keep the notation from the proof of Propo-sition 12, except redefine the quantities (cid:101) δ and (cid:101) (cid:15) to be (cid:101) δ = (cid:2)(cid:101) L + 1 + σ − rs − ( L − δ ( r )) (cid:3) + , (cid:101) (cid:15) = (cid:2) (cid:102) M + 1 + τ − rs ( M − (cid:15) ( r )) (cid:3) + . Arguing as in Step 1 of that proof, we have0 < (cid:101) α < (cid:98) (cid:101) L (cid:99) , < (cid:101) β < (cid:98) (cid:102) M (cid:99) , by taking r large enough, and also0 ≤ (cid:101) δ < (cid:98) (cid:101) L (cid:99) − (cid:101) α, ≤ (cid:101) (cid:15) < (cid:98) (cid:102) M (cid:99) − (cid:101) β. Step 2 — Estimating the counting function. Recall F represents the antiderivativeof f , defined in (18). Applying part (a) of [5, Proposition 6.1] to the curve (cid:101) Γ andusing the relationships between the unshifted and shifted quantities as in the proofof Proposition 12, we get (cid:12)(cid:12)(cid:12) (cid:101) N − r Area(Γ) + r (cid:16) F (cid:0) (1 + σ ) s/r (cid:1) + G (cid:0) (1 + τ ) s − /r (cid:1)(cid:17) + r (cid:16) sf (cid:0) (1 + σ ) s/r (cid:1) + s − g (cid:0) (1 + τ ) s − /r ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) ≤ r / (cid:16) (cid:90) Lα f (cid:48)(cid:48) ( x ) / d x + (cid:90) Mβ g (cid:48)(cid:48) ( y ) / d y (cid:17) + 175 r / (cid:16) s − / f (cid:48)(cid:48) (cid:0) L − δ ( r ) (cid:1) / + s / g (cid:48)(cid:48) (cid:0) M − (cid:15) ( r ) (cid:1) / (cid:17) + 700 r / (cid:16) l − (cid:88) i =0 s − / f (cid:48)(cid:48) ( α i ) / + m − (cid:88) j =0 s / g (cid:48)(cid:48) ( β j ) / (cid:17) + 14 (cid:16) l − (cid:88) i =0 s | f (cid:48) ( α i ) | + m − (cid:88) j =0 s − | g (cid:48) ( β j ) | (cid:17) + 12 r (cid:0) s − δ ( r ) + s(cid:15) ( r ) (cid:1) + l + m + 12 (1 + σ ) + 12 (1 + τ ) + (1 + σ )(1 + τ ) + 5+ rs − g ((1 + τ ) /rs ) − (1 + σ ) rsf ((1 + σ ) s/r ) − (1 + τ ) + rsf ((1 + σ ) s/r ) − (1 + τ ) rs − g ((1 + τ ) /rs ) − (1 + σ ) , (29)where we estimated the term involving (cid:101) f (cid:48)(cid:48) ( (cid:101) L − (cid:101) δ ) − / as follows. One has (cid:101) f (cid:48)(cid:48) ( (cid:101) L − (cid:101) δ ) = r − s f (cid:48)(cid:48) ( z ) where z = r − s ( (cid:101) L − (cid:101) δ + 1 + σ ) ≤ L − δ ( r ) , and so by monotonicity of f (cid:48)(cid:48) on each subinterval of the partition (as assumed inWeaker Convex Condition 2.6) one concludes (cid:101) f (cid:48)(cid:48) ( (cid:101) L − (cid:101) δ ) ≥ r − s min { f (cid:48)(cid:48) (cid:0) L − δ ( r ) (cid:1) , f (cid:48)(cid:48) (cid:0) α (cid:1) , . . . , f (cid:48)(cid:48) (cid:0) α l − (cid:1) } . Thus the term involving (cid:101) f (cid:48)(cid:48) ( (cid:101) L − (cid:101) δ ) − / can be estimated by the sum of terms involving f (cid:48)(cid:48) ( L − δ ( r )) − / and f (cid:48)(cid:48) ( α i ) − / .The right side of (29) has the form O ( r − e ), by arguing directly with s + s − = O (1)and the assumptions in Weaker Convex Condition 2.6, and estimating the last twoterms in (29) by rs − g ((1 + τ ) /rs ) − (1 + σ ) rsf ((1 + σ ) s/r ) − (1 + τ ) = s − L − o (1) sM − o (1) = O (1)and similarly with f and g interchanged.Step 3 — Understanding the left side of inequality (29). The terms on the left of (29)are dealt with in the same manner as in Step 3 of Proposition 12, except replacingLemma 13 with the last assumption in Weaker Convex Condition 2.6, as follows.Substituting x = (1+ σ ) s/r into f ( x ) = M + O ( x a ) and into F ( x ) = M x + O ( x a )gives rsf ((1 + σ ) s/r ) = rsM + O ( r − a ) ,r F ((1 + σ ) s/r ) = rs (1 + σ ) M + O ( r − a ) , HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 21 since s + s − = O (1). One argues similarly for g and G . Thus we have finished theproof under the assumption that r Γ( s ) passes through no lattice points.Step 4 — Finishing the proof. Now drop the assumption that r Γ( s ) passes throughno lattice points. Notice the counting function N ( r, s ) is increasing in the r -variable.Fix the r and s values, and modify the functions δ ( · ) and (cid:15) ( · ) to be continuous at r . For sufficiently small η > N ( r + η, s ) = N ( r, s ), because the r -variablewould have to increase by some positive amount for the curve r Γ( s ) to reach any newlattice points. Since no lattice points lie on the curve ( r + η )Γ( s ), Steps 1–3 aboveapply to that curve. Hence by continuity as η → 0, the conclusion of the propositionholds also for r Γ( s ). (cid:3) Lower bound on the counting function for decreasing ΓWe need a rough lower bound on the counting function, in order to prove bound-edness of the maximizing set in Theorem 1. Assume the curve Γ is strictly decreasingin the first quadrant, and has x - and y -intercepts at L and M . The intercepts neednot be equal, in the next lemma. Lemma 18 (Rough lower bound for decreasing curve) . The number N ( r, s ) of shiftedlattice points lying inside r Γ( s ) satisfies N ( r, s ) ≥ r Area(Γ) − r (cid:0) s − (1 + τ ) L + s (1 + σ ) M (cid:1) , r, s > . (30) Proof. We split the proof into two cases, and later rescale to handle the general curve.Write N for the number of shifted lattice points under Γ.Case I: The point (1 + σ, τ ) lies outside the curve Γ, and so N = 0. Then therectangles with vertices (0 , , ( L, , ( L, τ ) , (0 , τ ) and (0 , , (1 + σ, , (1 + σ, M ) , (0 , M ) cover Γ since the curve is decreasing, and so by comparing areas onehas N + (1 + τ ) L + (1 + σ ) M ≥ Area(Γ) . (31)Case II: The point (1 + σ, τ ) lies inside the curve. We shift the origin to (cid:101) O = (1 + σ, τ ) and draw new axes, denoting the x - and y -intercepts on the newaxes by (cid:101) L and (cid:102) M ; see Figure 7. The part of Γ lying in the new first quadrant is (cid:101) Γ.Each lattice point corresponds to a square whose lower left vertex sits at that point.These squares cover (cid:101) Γ since the curve is strictly decreasing. The remaining area underΓ is covered by the two rectangles described in Case I. The sum of the areas of thesquares and rectangles must exceed the area under Γ, and so (31) holds once again.To complete the proof, simply replace the curve Γ with r Γ( s ), meaning that in (31)we replace N , L , M with N ( r, s ), rs − L , rsM respectively. The lemma follows. (cid:3) Proof of Theorem 1 We prove the theorem in two parts: first for concave curves, and then for convexcurves. When Γ is concave, we will utilize the bound on S ( r ) in Lemma 9 and thetwo-term upper bound on the counting function in Proposition 10, along with the Figure 7. Decreasing curve Γ enclosing positive integer lattice pointsshifted by amount ( σ, τ ) = (0 . , − . σ, τ ,obtaining a new origin (cid:101) O , with (cid:101) L and (cid:102) M being the new x - and y -intercepts. The lattice point count equals the area of the squares, asused in proving Lemma 18.improved upper bound in Corollary 11 and the rough lower bound on the countingfunction in Lemma 18.Recall the intercepts are assumed equal ( L = M ) in this theorem. Part 1: Γ is concave and Parameter Assumption 2.1 holds. The proof hastwo steps. Step 1 shows S ( r ) is bounded above and below away from 0, for large r .Step 2 uses this boundedness to improve the asymptotic bound on S ( r ), revealingthat it depends only on σ and τ and not the curve Γ.Step 1. Take s ∈ S ( r ) and suppose r ≥ (2 + σ + τ ) /L . Then Lemma 9 says s ≤ rL/ (1 + σ ), so that r ≥ (1 + σ ) sL ≥ (1 − σ − ) sL . If s ≥ N ( r, s ) ≤ r Area(Γ) − C rs + σ − τ − . Parameter Assumption 2.1 guarantees here that C > s = 1” says N ( r, ≥ r Area(Γ) − (2 + σ + τ ) Lr. (32)Since s ∈ S ( r ) is a maximizing value, one has N ( r, s ) ≥ N ( r, s ≤ (2 + σ + τ ) LC + σ − τ − L (2 + σ + τ ) C when r ≥ (2 + σ + τ ) /L and s ≥ 1. Thus S ( r ) is bounded above for all large r .Similarly if s ∈ S ( r ) then s − is bounded above, by interchanging the roles of thehorizontal and vertical axes in the argument above. Thus the set S ( r ) is boundedbelow away from 0, for large r . HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 23 Step 2. The number s = lim sup s ∈ S ( r ) ,r →∞ s is finite and positive by Step 1. Combining the inequality N ( r, s ) ≥ N ( r, 1) withestimate (32) and Corollary 11 (which relies on the boundedness of S ( r )) we obtain( σ + 1 / s − (2 + σ + τ ) s + τ ≤ r → ∞ . Notice σ + 1 / > s isbounded above by the larger root of the quadratic; that is, s ≤ B ( σ, τ ) = 2 + σ + τ + (cid:112) (2 + σ + τ ) − σ + 1 / τ σ + 1 / . Similarly lim sup r →∞ s − ≤ B ( τ, σ ), by interchanging the roles of the axes. The proofof Theorem 1 is complete, in the concave case. Part 2: Γ is convex and Parameter Assumption 2.2 holds. Take s ∈ S ( r ) and suppose r satisfies (20), recalling there that µ f ( σ ) and µ g ( τ ) arepositive by Parameter Assumption 2.2. Now proceed as in Part 1 of the proof, simplyreplacing Lemma 9, Proposition 10 and Corollary 11 with Lemma 14, Proposition 15and Corollary 16, respectively.7. Proof of Theorem 5 Recall the intercepts are equal, L = M , in this theorem.The optimal stretch parameters are bounded above and bounded below away from0 as r → ∞ , by Theorem 1. (It suffices to use the curve-dependent bound from Step 1of that proof; we do not need the curve-independent bound B ( σ, τ ) from Step 2.)Hence by Proposition 12 (if Γ is concave) or Proposition 17 (if Γ is convex), N ( r, s ) = r Area(Γ) − rL (cid:0) s − ( τ + 1 / 2) + s ( σ + 1 / (cid:1) + O ( r − E ) (33)when s ∈ S ( r ); this estimate holds also when s > s ∈ S ( r ) and s ∗ = (cid:112) ( τ + 1 / / ( σ + 1 / 2) we have N ( r, s ) ≤ r Area(Γ) − rL (cid:0) s − ( τ + 1 / 2) + s ( σ + 1 / (cid:1) + O ( r − E ) ,N ( r, s ∗ ) ≥ r Area(Γ) − rL (cid:112) ( τ + 1 / σ + 1 / 2) + O ( r − E ) , as r → ∞ . Notice N ( r, s ∗ ) ≤ N ( r, s ) because s ∈ S ( r ) is a maximizing value, and so s − ( τ + 1 / 2) + s ( σ + 1 / ≤ (cid:112) ( τ + 1 / σ + 1 / 2) + O ( r − E ) . (34)Therefore s = s ∗ + O ( r −E ), by Lemma 19 below with a = τ + 1 / , b = σ + 1 / s ∈ S ( r ) one has2 (cid:112) ( τ + 1 / σ + 1 / ≤ s − ( τ +1 / s ( σ +1 / ≤ (cid:112) ( τ + 1 / σ + 1 / O ( r − E )by the arithmetic–geometric mean inequality and (34). Multiplying by rL and sub-stituting into (33) gives the asymptotic formula (9). Lemma 19. When a, b, s > and ≤ t ≤ √ ab , s − a + sb ≤ √ ab + t = ⇒ (cid:12)(cid:12) s − (cid:112) a/b (cid:12)(cid:12) ≤ ab ) / b √ t. Proof. By taking the square root on both sides of the inequality(( s − a ) / − ( sb ) / ) = s − a + sb − √ ab ≤ t and then using that the number ( ab ) / lies between ( s − a ) / and ( sb ) / (because itis their geometric mean), we find | ( ab ) / − ( sb ) / | ≤ t / . Hence ( ab ) / − t / ≤ ( sb ) / ≤ ( ab ) / + t / . Squaring and using that t ≤ ( ab ) / t / (when t ≤ √ ab ) proves the lemma. (cid:3) Proof of Theorem 7 and Proposition 8Proof of Theorem 7. Fix σ ∈ ( − , 0) and τ > − 1. Since 0 < σ < 1, we maychoose m ∈ N large enough that(1 + σ ) m < m + 1 . (35)Defining φ ( x ) = 1 − x m for 0 ≤ x ≤ 1, one checks φ (1 + σ ) > area under graph of φ .Thus one may choose 0 < δ < f ( x ) = 1 − δx − (1 − δ ) x m , ≤ x ≤ , satisfies f (1 + σ ) > area under graph of f . Observe f is smooth and strictly decreasing, with f (cid:48)(cid:48) < , g satisfies the same conditions.The curve r Γ( r ) is the graph of r f ( x ) for 0 ≤ x ≤ 1. This curve contains only thefirst column of shifted lattice points (the points with x -coordinate 1 + σ ), and so N ( r, r ) = (cid:98) r f (1 + σ ) − τ (cid:99)≥ r f (1 + σ ) − τ − . Now fix 0 < (cid:15) < 1. If s ∈ [ r (cid:15) − , r − (cid:15) ] then s + s − = O ( r − (cid:15) ), and so Proposition 12with q = 1 − (cid:15) and L = M = 1 gives that N ( r, s ) = r Area(Γ) − r (cid:0) s ( σ + 1 / 2) + s − ( τ + 1 / (cid:1) + O ( r − (cid:15)/ )= r Area(Γ) + o ( r ) . Since Area(Γ) < f (1 + σ ), we conclude that for all large r , N ( r, s ) < N ( r, r )and so s / ∈ S ( r ), which proves the theorem. HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 25 Figure 8. Maximizing s -values for the number of lattice points inthe 2-ellipse. (a) Left figure: positive shift σ = 1 , τ = 3. The plotshows log(sup S ( r ) − s ∗ ) versus log r . The line − / r indicates theguaranteed convergence rate in Theorem 2. (b) Right figure: negativeshifts σ = τ = − / 5. The plot shows log(sup S ( r )) and log(inf S ( r ))versus log r . Linear fitting gives log sup S ( r ) (cid:39) . 982 log r +0 . s (cid:38) r − (cid:15) proved in Proposition 8. Inboth plots, the r -values are multiples of √ / 10, an irrational numberchosen in the hope of exhibiting generic behavior. Proof of Proposition 8. By symmetry, we may suppose σ ≤ − / f ( x ) = √ − x . The only point to check in the proof is that f (1 + σ ) > Area(Γ)when − < σ ≤ − / 5, which reduces to the fact that 4 / > π/ Numerical examples, and conjectures for triangles ( p = 1)Figure 8(a) illustrates the convergence of s ∈ S ( r ) to s ∗ , when Γ is a quarter circleand the shifts are positive. The convergence is erratic, while still obeying the decayrate O ( r − / ) as promised by Theorem 2. Figure 8(b) shows the degeneration thatcan occur when the shifts are negative, as explained in Proposition 8.Quite different behavior occurs when Γ is a straight line with slope − 1, in otherwords, when the curve is the 1-ellipse described by f ( x ) = 1 − x , which is notcovered by our results in Example 6. Here N ( r, s ) counts the shifted lattice pointsinside the right triangle with vertices at ( r/s, , (0 , rs ) and the origin. Theorem 1insures the maximizing set S ( r ) is bounded above and below, being contained in (cid:2) B ( τ, σ ) − − ε, B ( σ, τ ) + ε (cid:3) for all large r . This boundedness depends on ParameterAssumption 2.1 holding, which in this case says(2 − max( σ − , τ − ))(1 − σ − − τ − ) > . In particular, S ( r ) is bounded for the 1-ellipse if σ = τ > − . S ( r ) does not apply, though, to the 1-ellipse.The numerical plots in Figure 9 suggest S ( r ) might not converge, and might insteadcluster at many different heights. Are those heights determined by a number theoreticproperty of some kind? (Such behavior would be particularly interesting when theshifts are σ = τ = − / 2, since those shifted lattice points correspond to energy levels Figure 9. Maximizing s -values for the number of lattice points in the1-ellipse (that is, the right triangle). The upper plots show log sup S ( r )versus r and the lower plots are log inf S ( r ) versus r . The figure onthe left is for shift parameters σ = τ = − / 2, which corresponds tocounting eigenvalues of the harmonic oscillator (Section 10). The mid-dle figure has σ = τ = 0, and the figure on the right has σ = τ = 4.Notice the optimal stretch parameters are bounded in a narrower andnarrower band as the shift parameters increase.of harmonic oscillators in 2-dimensions, as explained in the next section.) For a moredetailed discussion and precise conjecture on this open problem for p = 1 in theunshifted case, see our work in [18, Section 9] and the partial results of Marshall andSteinerberger [21].The numerical method that generated the figures is described in [18, Section 9] for p = 1. It adapts easily to handle other values of p , in particular p = 2 (the circle),and the code is available in [19, Appendix B].10. Future directions — optimal quantum oscillatorsLiterature on spectral minimization. Antunes and Freitas [2] investigated theproblem of maximizing the number of first-quadrant lattice points in ellipses withfixed area, and showed that optimal ellipses must approach a circle as the radiusapproaches infinity. In terms of eigenvalues of the Dirichlet Laplacian on rectangleshaving fixed area, their result says that the rectangle minimizing the n -th eigenvaluemust approach a square as n → ∞ . Their intuition is that high eigenvalues shouldbe asymptotically minimal for the “most symmetrical” domain.Besides Antunes and Freitas’s work on eigenvalue minimization [1, 2, 3, 10], wemention that van den Berg and Gittins [7] showed the cube is asymptotically minimalin 3-dimensions as n → ∞ , while Gittins and Larson [11] handle all dimensions ≥ ≥ / 2. Eigenvalue minimizing domains havebeen studied numerically by Oudet [22], Antunes and Freitas [2], and Antunes andOudet [4], [12, Chapter 11]. Incidentally, Colbois and El Soufi [9, Corollary 2.2]proved subadditivity of n (cid:55)→ λ ∗ n (the minimal value of the n -eigenvalue), from which HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 27 it follows that the the famous P´olya conjecture λ n ≥ πn/ Area in 2-dimensions wouldbe a corollary of the conjecture that the eigenvalue minimizing domain approaches adisk as n → ∞ .A different way of extending the work of Antunes and Freitas is to investigatelattice point counting inside more general curves, not just ellipses. Laugesen and Liu[18] and Laugesen and Ariturk [5] showed this can be done for p -ellipses with p (cid:54) = 1,and for more general concave and convex curves in the first quadrant too. Marshall[20] has extended the results to strongly convex domains in all dimensions, usingsomewhat different methods. For more on the literature see [5]. Spectral application of Proposition 8. This paper sheds new light on the rect-angular result of Antunes and Freitas. Consider the family of rectangles defined by[0 , πs − ] × [0 , πs ]for s > 0. The “even–even” eigenfunctions (that are symmetrical with respect to thetwo axes through the center) have the form u = sin (cid:0) s ( j − / x (cid:1) sin (cid:0) s − ( k − / y (cid:1) with corresponding eigenvalues λ = (cid:16) s ( j − 12 ) (cid:17) + (cid:16) s − ( k − 12 ) (cid:17) , for j, k ≥ 1. These even–even eigenvalues have counting function { λ ≤ r } = number of points in the shifted lattice ( N − / × ( N − / 2) lyinginside or on the ellipse ( sx ) + ( s − y ) ≤ r = N ( r, s )where the shift parameters are σ = τ = − / S ( r ) of s -values that maximize the counting functiondoes not approach 1 as r → ∞ . Instead, the maximizing s -values approach 0 or ∞ . Thus the even–even symmetry class of eigenvalues on the rectangle behaves quitedifferently from the full collection of eigenvalues studied by Antunes and Freitas. Theasymptotically optimal rectangle for maximizing the counting function as r → ∞ (orequivalently, minimizing the n -th eigenvalue as n → ∞ ) is not the square but ratherthe degenerate rectangle. Open problem for harmonic oscillators. A quantum analogue of the Antunes–Freitas theorem for rectangles would be to minimize the n -th energy level among thefollowing family of harmonic oscillators. For each s > 0, consider − ∆ u + 14 (cid:0) ( sx ) + ( s − y ) (cid:1) u = Eu, x, y ∈ R , (36)with boundary condition u → | ( x, y ) | → ∞ . Write s n for an s -value thatminimizes the n -th eigenvalue E n . By analogy with Antunes and Freitas’s theorem for Dirichlet rectangles, one might conjecture that s n → n → ∞ . In fact, thebehavior is quite different, as we now explain.Let us translate the harmonic oscillator problem into a shifted lattice point countingproblem. The 1-dimensional oscillator equation − u (cid:48)(cid:48) + x u = Eu has eigenvalues E = j − / j = 1 , , , . . . . . By separating variables and rescaling, one finds thatequation (36) has spectrum { E n } = { s ( j − / 2) + s − ( k − / 2) : j, k = 1 , , , . . . } . Hence the number of harmonic oscillator eigenvalues less than or equal to r equals thenumber of points in the shifted lattice ( N − / × ( N − / 2) lying below the straightline sx + s − y = r , which is given by our counting function N ( r, s ) where Γ is thestraight line y = 1 − x (the 1-ellipse) and the shift parameters are σ = τ = − / 2. Tominimize the eigenvalues we should maximize the counting function.The numerical evidence in the left part of Figure 9 suggests that the s -valuesmaximizing the counting function N ( r, s ) do not converge to 1 as r → ∞ . Rather,the optimal s -values seem to cluster at various heights. (For a precise such clusteringconjecture in the unshifted case, see [18, Section 9].) Thus the family of harmonicoscillators exhibits strikingly different spectral behavior from the family of Dirichletrectangles. Interpolating family of Schr¨odinger potentials. The family of Schr¨odinger po-tentials | sx | q + | s − y | q , where 2 < q < ∞ and s > 0, interpolates between the harmonicoscillator ( q = 2) and the infinite potential well ( q = ∞ ) that corresponds to theDirichlet Laplacian on a rectangular domain. We conjecture that when 2 < q < ∞ ,the set S ( r ) of values maximizing the eigenvalue counting function will converge to1 as r → ∞ . This conjecture would provide a 1-parameter family of quantum os-cillators for which the analogue of the Antunes–Freitas theorem holds true, with thefamily terminating in an exceptional endpoint case: the harmonic oscillator.The difficulty is that the eigenvalues of the 1-dimensional oscillator with potential | x | q do not grow at a precisely regular rate. Hence to tackle the conjecture, one willneed to extend the current paper from shifted lattices, where each row and columnof the lattice is translated by the same amount, and find a way to handle deformed lattices, where the amount of translation varies with the rows and columns. Thischallenge remains for the future. Acknowledgments This research was supported by grants from the Simons Foundation ( HIFTED LATTICES AND ASYMPTOTICALLY OPTIMAL ELLIPSES 29 References [1] P. R. S. Antunes and P. Freitas. Numerical optimization of low eigenvalues of the Dirichlet andNeumann Laplacians. J. Optim. Theory Appl. 154 (2012), 235–257.[2] P. R. S. Antunes and P. Freitas. Optimal spectral rectangles and lattice ellipses. Proc. R. Soc.Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2150, 20120492, 15 pp.[3] P. R. S. Antunes and P. Freitas. Optimisation of eigenvalues of the Dirichlet Laplacian with asurface area restriction. Appl. Math. Optim. 73 (2016), no. 2, 313–328.[4] P. R. S. Antunes and ´E. Oudet. Numerical minimization of Dirichlet–Laplacian eigenvalues offour-dimensional geometries. SIAM J. Sci. Comput., to appear.[5] S. Ariturk and R. S. Laugesen. Optimal stretching for lattice points under convex curves. Port.Math., to appear. ArXiv:1701.03217[6] M. van den Berg, D. Bucur and K. Gittins. Maximizing Neumann eigenvalues on rectangles. Bull. Lond. Math. Soc. 48 (2016), no. 5, 877–894.[7] M. van den Berg and K. Gittins. Minimising Dirichlet eigenvalues on cuboids of unit measure. Mathematika 63 (2017), no. 2, 469–482.[8] D. Bucur and P. Freitas. Asymptotic behaviour of optimal spectral planar domains with fixedperimeter. J. Math. Phys. 54 (2013), no. 5, 053504.[9] B. Colbois and A. El Soufi. Extremal eigenvalues of the Laplacian on Euclidean domains andclosed surfaces. Math. Z. 278 (2014), 529–549.[10] P. Freitas. Asymptotic behaviour of extremal averages of Laplacian eigenvalues. J. Stat. Phys.167 (2017), no. 6, 1511–1518.[11] K. Gittins and S. Larson Asymptotic behaviour of cuboids optimising Laplacian eigenvalues ArXiv:1703.10249[12] A. Henrot, ed. Shape Optimization and Spectral Theory. De Gruyter Open, to appear, 2017.[13] M. N. Huxley. Area, Lattice Points, and Exponential Sums. London Mathematical SocietyMonographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York,1996, Oxford Science Publications.[14] M. N. Huxley. Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87 (2003),591–609.[15] E. Kr¨atzel. Analytische Funktionen in der Zahlentheorie. Teubner–Texte zur Mathematik, 139.B. G. Teubner, Stuttgart, 2000. 288 pp.[16] E. Kr¨atzel. Lattice points in planar convex domains. Monatsh. Math. 143 (2004), 145–162.[17] S. Larson. Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convexdomains. ArXiv:1611.05680[18] R. S. Laugesen and S. Liu. Optimal stretching for lattice points and eigenvalues. Asymptotically optimal shapes for counting lattice points and eigenvalues. Ph.D. disser-tation, University of Illinois, Urbana–Champaign, 2017.[20] N. F. Marshall. Stretching convex domains to capture many lattice points. ArXiv:1707.00682.[21] N. F. Marshall and S. Steinerberger. Triangles capturing many lattice points . ArXiv:1706.04170.[22] ´E. Oudet. Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10 (2004), 315–330. Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A. E-mail address ::