Geometric Phase Integrals and Irrationality Tests
aa r X i v : . [ m a t h . N T ] D ec Geometric Phase Integrals andIrrationality Tests
Domenico Napoletani ∗ , and Daniele C. Struppa † Abstract
Let F ( x ) be an analytical, real valued function defined on a com-pact domain B ⊂ R . We prove that the problem of establishing theirrationality of F ( x ) evaluated at x ∈ B can be stated with respectto the convergence of the phase of a suitable integral I ( h ), defined onan open, bounded domain, for h that goes to infinity. This is derivedas a consequence of a similar equivalence, that establishes the exis-tence of isolated solutions of systems equations of analytical functionson compact real domains in R p , if and only if the phase of a suitable“geometric” complex phase integral I ( h ) converges for h → ∞ . Wefinally highlight how the method can be easily adapted to be rele-vant for the study of the existence of rational or integer points oncurves in bounded domains, and we sketch some potential theoreticaldevelopments of the method. ∗ Institute for Quantum Studies, Chapman University, Orange, CA, 92866.Email:[email protected] † Schmid College of Science and Technology, Chapman University, Orange, CA 92866.Email: [email protected] Real Geometry and Irrationality
Real geometry, and especially real algebraic geometry, has developed rela-tively late its own techniques [5], perhaps due to the great success of alge-braic geometry theories over complex fields. However, this has led to a lackof tight bounds on the structure of solutions of systems of equations overthe real numbers, and, even with the extensive recent development of realalgebraic geometry and its relations to the theory of computation ( see forexample [4]), a general tool that can encompass problems on a very largeclass of functions is lacking.In this paper we show that the existence of isolated solutions of a systemof analytical equations F ( x ) = 0 over a compact B in R p is equivalent,under suitable conditions, to the existence of the limiting phase of a complexphase integral I F ( h ) for h that goes to infinity. This result allows a plethoraof analytical techniques for the asymptotic and non-perturbative study ofcomplex phase integrals to become relevant for real geometry, providing, insome sense, an indirect, but tightly tailored complexification of real geometry.We then approach another, apparently unrelated question: establishingthe irrationality of special numbers. Geometry has always been deeply in-tertwined with the problem of establishing the irrationality of numbers, but,while the more specific problem of establishing linear and algebraic inde-pendence of several point evaluations of special functions has a rich modernhistory [2], establishing directly the irrationality of series, and of pointwiseevaluation of general functions, has been, mostly, the domain of ad-hoc meth-ods (see for example [3, 8]).In Section 3 we suggest that, for real irrational numbers, another view-point is available, that transforms the problem of the irrationality of F ( x ) = α into the geometric problem of finding zeros of a systems of equations ona four dimensional open, bounded domain. This problem is then phrased interms of the phase integral method we developed for real geometry in Sec-tion 2, offering a new perspective on some old problems. We conclude the2aper by suggesting how the main results of Section 3 can be adapted todiophantine geometry. Given an analytical vectorial function F ( x ) = [ F ( x ) , ..., F p ( x )] defined ona compact set B ⊂ R p , consider the associated norm function L ( x ) = P pi =1 F i ( x ) . Then L ( x ) = 0 clearly implies F i ( x ) = 0 for all i . More-over, every point such that L ( x ) = 0 is a critical point of L ( x ), since ∂L ( x ) ∂x i = P pj =1 F j ( x ) ∂F j ( x ) ∂x i and setting F j ( x ) = 0 for all j ’s gives ∂L ( x ) ∂x i = 0.The relations between critical points of L ( x ) and solutions of the systemof equations F ( x ) = 0 can be made more compelling, by building a suit-able phase integral whose asymptotic behavior depends on the existence ofsolutions to the system itself. Indeed the following theorem holds: Theorem 2.1.
Let F ( x ) = [ F ( x ) , ..., F p ( x )] be an analytical, vectorial func-tion defined on a compact domain B ⊂ R p , and let L ( x ) = P pi =1 F i ( x ) haveonly isolated critical points in B . Consider the integral I ( h ) = Z A Z B e ihL ( x ) y dxdy, y ∈ A ⊂ R ,
6∈ A , x ∈ B ⊂ R p , (1) and denote by φ ( I ( h )) the phase of I ( h ) , then the system F ( x ) = 0 has asolution in B if and only if the phase φ ( I ( h )) has a limit for h going toinfinity.Proof. The integration in x in the integral in Eq. 1 can be written, for h → ∞ , with respect to the critical points of L ( x ) in B , using standardstationary phase approximation methods [10, 7], in this paper we will refermostly to [10] for the necessary background material.We can consider separately the critical points such that L ( x ) = 0, and3hose for which L ( x ) = 0,and we have:lim h →∞ I ( h ) = Z y ∈A X L ( x i )=0 ( 2 πh ) p y p (det H ( x i )) / e i π σ i dy + Z y ∈A X L ( x j ) =0 ( 2 πh ) p y p (det H ( x j )) / e ihL ( x j ) y + i π σ j dy (2)with H ( x ∗ ) the Hessian matrix of L ( x ) evaluated at x ∗ , and σ ∗ the signatureof H ( x ∗ ).We are assuming here that there is at least one critical point with Hessiandifferent from zero, very similar arguments to those we present here can bededuced without this restriction, at the cost of a more complicated argumentthat depends on higher derivatives of L (that exist, since L is analytical in B ). The restricted (but generic) setting with at least one critical point withHessian different from zero is sufficient to prove our main result in Section 3.Since the function L is analytical, so is the system of equations whose so-lution defines critical points, and if we assume all such solutions are isolated,they are in finite number over a compact set (see for example [9], page 180).And we can further simplify the representation of I ( h ) and in particular, inthe limit of large h ,lim h →∞ I ( h ) = lim h →∞ Z y ∈A X L ( x i )=0 ( 2 πh ) p y p (det H ( x i )) / e i π σ i dy =lim h →∞ X L ( x i )=0 ( 2 πh ) p H ( x i )) / e i π σ i S (3)where S = R A y p dy , and, being the last term a finite sum, and factoring out h p/ , the phase of I ( h ) is shown to be independent of h and dependent onlyon the critical points x i ’s, or more exactly, on the signatures σ i .Let us now analyze the second portion of the sum in the right hand side4f Eq. 2: I ( h ) = Z y ∈A X L ( x j ) =0 ( 2 πh ) p y p (det H ( x j )) / e ihL ( x j ) y + i π σ j dy (4)We note first of all that each integral Z y ∈A ( 2 πh ) p y p (det H ( x j )) / e ihL ( x j ) y + i π σ j dy (5)can be written as( 2 πh ) p H ( x j )) / e i π σ j Z y ∈A y p e ihL ( x j ) y dy (6)and it is therefore a phase integral in y computed over an interval that doesnot include a critical point ( y = 0). Such integral decreases at least like O ( hL ( x j ) ), the leading contribution from the boundary points of A ([10],page 488; [7], page 52). And thereforelim h →∞ I ( h ) = lim h →∞ (2 π ) p X L ( x j ) =0 H ( x j )) / e i π σ j O ( 1 h p/ L ( x j ) ) (7)Recall we are in the generic case where there is at least one point x j with det H ( x j ) = 0, and we know there are finitely many critical points, andtherefore also finitely many critical points for which L ( x j ) = 0. This lastobservation allows us to conclude that all the values L ( x j ) can be boundedaway from 0, and the entire sum above can be estimated aslim h →∞ I ( h ) = O ( 1 h p/ ) (8)This is a negligible quantity with respect to I ( h ) ∼ h p/ . We can concludethat the limit for h → ∞ of I ( h ) = I ( h ) + I ( h ) has constant phase if L ( x ) = 0 for at least a specific x j . If there are no values for which L ( x ) = 0,5he phase will not converge, this is easy to see in the case we do have atleast a critical point x j with L ( x j ) = 0 and det H ( x j ) = 0, since in that casethe term e ihL ( x j ) y in I ( h ) will each continue to change phase as h goes toinfinity.Note that if the critical points such that L = 0 have det H = 0, wewould need to look at higher order asymptotic terms, but, since the numberof critical points is finite, we could still look at the highest order, dominantcritical points, whose phase is dependent on e ihL ( x j ) y ([10] page 483), and thisis one of the reasons we need to have, in the most general case, F analytical.Suppose instead that there are no critical points at all, then the integralin Eq. 1 is dominated by the evaluation of some derived phase integral onthe boundary of A × B , more particularly, it is true that (adapted from [10],page 488): I ( h ) ∼ − ih Z ∂ ( A × B ) Ge ihL ( x ) y da (9)where ∂ ( A × B ) is the boundary of
A × B , da is a suitable measure on theboundary, and G is a multiplier function dependent on L ( x ) y .Now, A × B is an hypercube, and a recursive application of the resultin Eq. 9, to lower and lower dimensional boundaries of its hyperfaces, willreduce the asymptotic evaluation of I ( h ) to a sum of suitable multiples ofevaluations of e ihL ( x ) y at the vertexes of the hypercube. None of these valuesis independent of h , since we assumed there are no critical points of L on A ×B , and therefore L ( x ) y = 0 everywhere. This implies that lim h →∞ φ ( I ( h ))does not exist when there are no critical points on A × B . Remark 2.2.
While the main thrust of this paper is the analysis of irra-tionality of the evaluation of analytical functions at a point, as it will beclear in the next section, we stress that the result of Theorem 2.1, in its Where successive multiplier functions G i will depend both on L y and G i − (seeagain [10] page 487-488). implicity, offers a potentially powerful new approach for problems in realgeometry, and in particular for the solution of problems in real algebraic ge-ometry. To this purpose, Theorem 2.1 would need to be suitably generalizedto the case F ( x ) = 0 has solutions of dimension bigger than zero, along thelines of the results on stationary phase asymptotic approximations on curvesdescribed in [10], page 459. Because of the property proven in Theorem 2.1 that the phase of I ( h ) inEq. 1 is constant in the limit of h large if and only if there is a solution for theequation F ( x ) = 0, we call the integrals in Eq. 1 geometric phase integrals .And we call L ( x ) the geometric Lagrangian associated to F ( x ) = 0. Weuse this terminology in analogy to the Lagrangian functions used in definingpath and field integrals [1], trusting that it will be suggestive of furthercrossfertilization of ideas and methods. In our main setting of the study ofirrationality of evaluation of functions, see for example the discussion at theend of Section 4. We will now apply this general setting to a more complex case that involvesinfinitely many critical points, but such that the relative contributions ofeach can be controlled.Suppose we want to know whether F ( x ) = α is irrational. The systemof equations F ( x ) − α = 0 , x ∈ [ x − δ, x + δ ] x − x = 0 , α ∈ F ([ x − δ, x + δ ])sin πm = sin πn = 0 , m, n ∈ (0 , αm − n = 0 (10)has a solution if and only if α is a rational number. We can adapt the7tationary phase integral analysis performed in Section 2, used to study ge-ometric problems, to be of relevance in this case. We build to this purposethe geometric Lagrangian function: L ( x, α, m, n ) = ( F ( x ) − α ) + ( x − x ) + sin πm + sin πn + ( αm − n ) (11)Again, L ( x, α, m, n ) = 0 if and only if the previous system has a zero solution,and we may ask whether the limit for h → ∞ of the phase of the followingintegral has any relation to the rationality of F ( x ) = α : I L ( h ) = Z y ∈A Z ω ∈ Ω δ e ihL ( ω ) y dωdy,
6∈ A (12)where ω = ( x, α, m, n ) and we denote by Ω δ the tensor product of the domainsallowed for each of the components of ω in Eq. 10.The main complication, with respect to the similar setting in Section 2,is the existence of infinitely many critical points, every time there is at leastone point such that L ( ω ) = 0. Consider the partial first derivatives of L ( ω ),a critical point of L ( ω ) has to satisfy: ∂L∂x = 2( F ( x ) − α ) dF ( x ) dx + 2( x − x ) = 0 ∂L∂α = − F ( x ) − α ) + 2( αm − n ) = 0 ∂L∂m = 2 sin πm cos πm ( − πm ) + 2( αm − n ) α = 0 ∂L∂n = 2 sin πn cos πn ( − πn ) − αm − n ) = 0 (13)We can see that if ω = ( x , α , m , n ) is a solution of L ( ω ) = 0, then it isalso a critical point of L . However, also ω i = ( x , α , m i , n i ) will be a zeroand a critical point of L , where m i = m i and n i = n i , i any integer (this canbe seen by simple substitution in αm − n = 0, assuming α m − n = 0).8ote that all critical points with L ( ω ) = 0 need to have x = x and α = α .To overcome this proliferation of critical points there are two main issuesto consider, the first is that our argument will work only in the limit of thedomain approaching the zero for variables m, n . Second, we need to controlthe decay of the Hessian in the asymptotic expression used to prove Theorem2.1.Regarding the first issue, we cut the domain of m and n as m ∈ [ M, n ∈ [ N,
1] with 0 < M, N < δ ( M, N ) = [ x − δ, x + δ ] × F ([ x − δ, x + δ ]) × [ M, × [ N, . (14)The main conclusion of our analysis can be stated as a theorem: Theorem 3.1.
Let F ( x ) be an analytical function in the interval [ x − δ, x + δ ] , with δ sufficiently small, and assume F ′ ( x ) = 0 . Consider the followingphase integral, the restriction of I L ( h ) to the domain Ω δ ( M, N ) : I L ( h, M, N ) = Z y ∈A Z ω ∈ Ω δ ( M,N ) e ihL ( ω ) y dωdy,
6∈ A (15) where L is defined in Eq. 11. Let φ ( I L ( h, M, N )) be the phase of I L ( h, M, N ) . F ( x ) = α is a rational number if and only if the following limit converges: lim M,N → lim h →∞ φ ( I L ( h, M, N )) . (16) Proof.
We start our proof with a simple analysis of the dimensionality, in x and α , of solutions of the first equation of the systems in Eq. 13, that definethe critical points. And we eventually prove that for δ small enough all criticalpoints in Ω δ are isolated. To achieve this goal, we note that, for δ sufficientlysmall we can control the norm of another function, ( F ( x ) − α ) − F ′ ( x )( x − x ),9n Ω δ , indeed we have | ( F ( x ) − α ) − F ′ ( x )( x − x ) | = | ( F ( x ) − ( F ( x ) + ǫ )) − ( F ′ ( x ) + ǫ )( x − x ) | = | ( F ( x ) − F ( x )) − F ′ ( x )( x − x ) − ǫ − ǫ ( x − x ) | ≤| ( F ( x ) − F ( x )) − F ′ ( x )( x − x ) | + | ǫ | + | ǫ ( x − x ) | ≤| ( F ( x ) − F ( x )) − F ′ ( x )( x − x ) | + | ǫ | + | ǫ δ | ≤| ǫ | + | ǫ | + | ǫ δ | (17)where we used the fact that the derivative of F ( x ) is well defined and con-tinuous in a neighborhood of x , and ǫ t , t = 1 , ,
3, can be made as smallas necessary choosing δ small enough. But we can interpret this result bysaying that the vectors ( F ( x ) − α, x − x ) and (1 , − F ′ ( x )) are almost or-thogonal for all ( x, α ) in Ω δ , with δ sufficiently small. Now the equation2( F ( x ) − α ) dF ( x ) dx + 2( x − x ) = 0 in Eq. 13 is equivalent to saying that( F ( x ) − α, x − x ) and ( F ′ ( x ) ,
1) are orthogonal, for some choice of ( x, α )in Ω δ . Together with the previous calculations, this implies, for two dimen-sional vectors, that ( F ′ ( x ) ,
1) and (1 , − F ′ ( x )) should be almost parallel, forsuch choice of ( x, α ), instead, these vectors are themselves orthogonal, andwe conclude there is no solution of 2( F ( x ) − α ) dF ( x ) dx + 2( x − x ) = 0, unless( F ( x ) − α, x − x ) = (0 , x = x and α = F ( x ). Note thatthis argument depends on the assumption F ′ ( x ) = 0 otherwise we wouldnot be able to infer α = F ( x ) from x = x , in the first equation of Eq. 13.We deduce moreover, from the whole set of equations in Eq. 13, thatcritical points with x = x and α = F ( x ), if they exists, are bound to have αm − n = 0, 2 sin πm cos πm ( − πm ) = 0, and 2 sin πn cos πn ( − πn ) = 0. Thereforethey are all isolated points, in finite number on all compacts Ω δ ( M, N ) andthey either satisfy sin πm = 0 and sin πn = 0 (and therefore L ( ω ) = 0), or theyare such that cos πm = 0 and/or cos πn = 0. Since critical points are isolatedand finitely many in Ω δ ( M, N ), for any 0 < M, N <
1, we are in the position10f applying Theorem 2.1 in the rest of the proof.The proof of the theorem then relies on the following estimate: suppose α is rational and that m , n are the largest values such that L ( x , α , m , n ) =0, thendet H ( x , α , m i , n i ) ∼ C i m (18)in the limit of i that goes to infinity, where m i = m i , n i = n i , i positiveinteger and C is a positive number bigger than 1. Indeed, remembering that,for critical points ω i = ( x , α , m i , n i ) with L ( ω i ) = 0, we have α m i − n i = 0,sin πm i = 0, sin πn i = 0 (and therefore cos πn i = 1, cos πm i = 1), we can writethe Hessian matrix of L ( ω ) evaluated at such critical points as: H ( ω i ) = F ′ ( x ) + 2 − F ′ ( x ) 0 0 − F ′ ( x ) 2 + 2 m i α m i − m i α m i π m i + 2 α − α − m i − α π n i + 2 . (19)Using again the fact that, for these critical points, α m i = n i , the evaluationof the determinant gives:det H ( ω i ) = (4 F ′ ( x ) + 4 m i + 4) (cid:0) π m i + α )( π α m i + 1) − α (cid:1) + 4( F ′ ( x ) + 2) α m i ( − π α m i − − F ′ ( x ) + 1) m i ( π m i + α ) . (20)where we did not fully simplify the expression to leave the reader with asense of its structure. Recalling m i = m i with i = 1 , , ... , if we let i → ∞ m i → H ( ω ) ∼ π m i π α m i = 16 π α i m (21)which is the estimate in Eq. 18, with C = π α . This being the case, we canbe assured that there is a i T such that for i > i T the Hessian H ( x , α , m i , n i )has nonzero (positive) determinant, and therefore the quadratic asymptoticapproximation used in Theorem 2.1 holds for all i > i T .Also, note that, for i < i T any critical point such that H ( x , α , m i , n i ) =0 will depend from h , in the asymptotic expansion, as h j +2 for some integer j > H ( x , α , m i , n i ) = 0 depend from h as h ([10], page 480). This implies thatwe can neglect critical points that have Hessian equal to zero, in the limitof h → ∞ , since the asymptotic relation in Eq. 18 assures us that there areinfinitely many dominant critical points with non-zero determinant of theHessian in Ω δ , and therefore at least one of them for M, N sufficiently small.Therefore we have: lim
M,N → lim h →∞ φ ( I L ( h, M, N )) =lim M,N → lim h →∞ Z y ∈ A X L ( ω i )=0det H ( ω i ) =0 ω i ∈ Ω δ ( M,N ) ( 2 πh ) y (det H ( ω i )) / e i π σ i (22)where we have used the results from Theorem 2.1, the fact that p = 4, andneglected already the (finitely many) critical point for which L ( ω ) = 0, orthose for which L ( ω i ) = 0 and det H ( ω i ) = 0.Consider now the partial sums: θ M,N = X L ( ω i )=0det H ( ω i ) =0 ω i ∈ Ω δ ( M,N ) (2 π ) (det H ( ω i )) / e i π σ i (23)12hen lim M,N → lim h →∞ φ ( I L ( h, M, N )) = lim M,N → lim h →∞ φ (cid:0) Z y ∈ A h y θ M,N dy ) (cid:1) =lim M,N → lim h →∞ φ (cid:0) h Sθ M,N (cid:1) = lim
M,N → φ ( θ M,N ) (24)where S = R y ∈A y dy . Now, because of the relation det H ( x , α , m i , n i ) ∼ C i m , for i → ∞ we can argue that the following series converges: θ = X L ( ω i )=0det H ( ω i ) =0 ω i ∈ Ω δ (2 π ) (det H ( ω i )) / e i π σ i (25)Indeed, the convergence of the this series can be reduced to the conver-gence of its absolute value X L ( ω i )=0det H ( ω i ) =0 ω i ∈ Ω δ (2 π ) (det H ( ω i )) / (26)and, by comparison with the convergent series P i i , the limit comparisontest of convergence gives us:lim i →∞ (2 π ) (det H ( ω i )) / (cid:14) i = lim i →∞ (2 π ) √ C ( i /m ) / (cid:14) i = (2 π ) m / √ C (27)Since the limit of the quotient above is nonzero, the series in Eq. 26converges, and θ in Eq. 25 is well defined. The convergence of the seriesdefining θ allows us one final limiting argument, i.e.,lim M,N → lim h →∞ φ ( I L ( h, M, N )) = lim M,N → φ ( θ M,N ) = φ ( θ ) . (28)And this last equality completes the proof of the Theorem.13 emark 3.2. The convergence of the series defining θ in Eq. 25 is intimatelyrelated to the estimate in Eq. 18. The existence of this estimate depend onthe fact that we use the equations sin πm = 0 , sin πn = 0 , on a bounded domain,to force rationality of F ( x ) = α (via the additional equation αm − n = 0 ).Such convergence would not hold if rationality was enforced via the equations sin πm = 0 , sin πn = 0 on an unbounded domain. Note also that the phaseintegral in Eq. 12 depends functionally on F ( x ) , so that the local behavior of F ( x ) for x ∼ x becomes relevant for the irrationality of F ( x ) = α . Remark 3.3.
Our choice of the dependence of the geometric Lagrangiansfrom variable y is not the only one that would establish the results in Theo-rems 2.1 and 3.1, even though it is probably the simplest. Alternatively, onecould look at the geometric Lagrangian L ( ω ) exp( y ) + y whose critical pointsare only those associated to L ( ω ) = 0 , removing the necessity of the carefulestimate of the contribution of critical points with L ( ω ) = 0 . However, thismore complicated geometric Lagrangian leads always to degenerate criticalpoints in the stationary phase asymptotic approximation and therefore to amore intricate proof of the two Theorems. There are several problems that could benefit from the application of The-orem 3.1, however, we formally write only one such application for a number,the gamma constant γ , whose irrationality is not known. The Digamma func-tion Ψ can be used to define the Euler-Mascheroni γ constant as Ψ(1) = − γ ,and since Ψ( x ) is analytical at x = 1, with Ψ ′ (1) = π = 0 we can statethe following Corollary to Theorem 3.1, where we assume δ has already beenchosen sufficiently small: Corollary 3.4.
Consider the geometric Lagrangian associated to the Digammafunction Ψ : L ( x, α, m, n ) = (Ψ( x ) − α ) + ( x − + sin πm + sin πn +( αm − n ) . (29)14 he Euler-Mascheroni constant γ is rational if and only if the following limitconverges lim M,N → lim h →∞ φ ( I L ( h, M, N )) . (30) The method we outlined in Section 3 is not restricted to the study of ratio-nality of functions evaluated at one point. Suppose we are interested in theproblem of finding whether F ( x ) = 0, x ∈ B has rational solutions, with B a compact domain. The method described in Section 3 will apply, using theGeometric Lagrangian: L ( x, m, n ) = F ( x ) + p X i =1 sin πm i + sin πn i + ( x i m i − n i ) (31)A full adaptation of Theorem 3.1 to this case requires a careful evaluation ofconvergence of multiple series associated to critical points with F ( x ) = 0, andthis is problematic if there are infinitely many rational solutions of F ( x ) = 0in B . However, the proof of Theorem 3.1 directly applies when F ( x ) = 0 hasfinitely many rational solutions on B , just by considering the finitely manyconvergent series of the type in Eq. 25, associated to each rational solution,if any. This implies that a slightly modified version of Theorem 3.1 holds forthe study of rational curves of genus bigger than 1 since such curves alwayshave at most finitely many rational points [6].Also the problem of determining whether F ( x ) = 0 has algebraic solutionsof degree K can also be stated in terms of phase integrals on geometric15agrangians of the type: L ( x, a, m, n ) = F ( x ) + p X i =1 g i ( x i ) + p X i =1 K X j =1 sin πm ij + sin πn ij + ( a ij m ij − n ij ) (32)where g i ( x i ) are polynomials of degree at most K , a ij are their (rational)coefficients and we denote by a the vector of all a ij . It is not clear howeverwhether the phase integral method can be adapted to discriminate algebraicnumbers, of any degree K , from transcendental numbers.Diophantine equations could be similarly approached. If we are interestedin the existence of integer solutions of F ( x ) = 0 on a bounded domain B ⊂ R p , the following geometric Lagrangian would be suitable: L ( x, m ) = F ( x ) + p X i =1 sin πm i + ( x i m i − (33)We have already stressed in Remark 3.2 the importance of keeping afunctional dependence of the phase integral from F ( x ). This dependence canbe used to say a little more about the structure of the geometric Lagrangiansdefined so far, as they can all be split into three components. Let’s focus forsimplicity on the geometric Lagrangian for the irrationality test in Section3 (the same arguments extend easily to the Lagrangians sketched in thissection).The geometric Lagrangian L ( x, α, m, n ) = ( F ( x ) − α ) + ( x − x ) +sin πm + sin πn + ( αm − n ) can be written as L ( x, α, m, n ) = L ( x, α ) + L ( m, n ) + L ( α, m, n ), now the functions L ( x, α ) = ( F ( x ) − α ) + ( x − x ) and L ( m, n ) = sin πm + sin πn can be seen as two distinct Lagrangians,each leading to geometric phase integrals whose phase is always convergent,while L ( α, m, n ) = ( αm − n ) can be seen as a “coupling Lagrangian” that16rovides the interaction between the first two Lagrangians.This discussion is inspired by the language of quantum field theory, whereinteraction among free fields is often mediated by only some of the terms inthe associated Lagrangian ([1], chapter 5). If we push this analogy even fur-ther, we can say that, for any small coupling parameter β > L β ( x, α, m, n ) = ( F ( x ) − α ) + ( x − x ) + sin πm + sin πn + β ( αm − n ) = L ( x, α ) + L ( m, n ) + L ,β ( α, m, n ) is just as suitable to study the irrational-ity of F ( x ) = α . For β very small, this modification allows to expandthe phase integral associated to L β in terms of powers of L ,β , since, for β sufficiently small, L ,β = βL = β ( αm − n ) will also be small on Ω δ . Moreparticularly, for β very small, and under the conditions of Theorem 3.1, wehave the following equalities: Z y ∈A Z ω ∈ Ω δ ( M,N ) e ihL β ( ω ) y dωdy = Z y ∈A Z ω ∈ Ω δ ( M,N ) e ih ( L ( ω )+ L ( ω )) y e ihL ,β ( ω ) y dωdy = Z y ∈A Z ω ∈ Ω δ ( M,N ) e ih ( L ( ω )+ L ( ω )) y ∞ X j =0 ( ihL ,β ( ω ) y ) j j ! dωdy. (34)And, being mindful of the contrasting tension between the requirement h →∞ and β →
0, the study of the convergence of the phase of the first in-tegral in Eq. 34 could be replaced by the study of the convergence of thephase of the following series of integrals, potentially allowing perturbativeand renormalization methods to be relevant here: ∞ X j =0 ( ihβ ) j j ! Z y ∈A Z ω ∈ Ω δ ( M,N ) L ( ω ) j y j e ih ( L ( ω )+ L ( ω )) y dωdy. (35)Not only, it is possible to construct an entire family of Lagrangians { L β } ,and study the structure of the “flow” of the associated phase integrals as β →
0. Note that, for any β = 0, if F ( x ) = α is irrational, there will17e no phase convergence of I ( h, M, N ) as defined in Theorem 3.1, but for β = 0 there will always be phase convergence since the Lagrangians L and L will be decoupled in that case, and there will always be solutions to theassociated geometric problem. So the problem of irrationality of F ( x ) = α can also be approached as an abrupt qualitative transition, at β = 0, of thestructure of the family of phase integrals associated to the Lagrangians { L β } ,again enriching irrationality problems with the methodologies that have beendeveloped to study phase transitions in physics.While this heuristic discussion is brief and very informal, it is included inthe paper to be suggestive of the significant conceptual shift that is possible,by using Theorems 2.1 and 3.1 as a starting point for a renewed study ofirrationality and real geometry. References [1] A. Altland, B. D. Simons,
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