Quantum invariants of hyperbolic knots and extreme values of trigonometric products
aa r X i v : . [ m a t h . N T ] J un Quantum invariants of hyperbolic knots andextreme values of trigonometric products
Christoph Aistleitner and Bence Borda
Graz University of TechnologySteyrergasse 30, 8010 Graz, AustriaEmail: [email protected] and [email protected]
Abstract
In this paper we study the relation between the function J , , whicharises from a quantum invariant of the figure eight knot, and Sudler’strigonometric product. We prove the convergence of suitably normalizedlogarithms of J , along points coming from continued fraction convergentsof a quadratic irrational, and we show that this asymptotics deviates fromthe universal limiting behavior that has been found by Bettin and Drappeauin the case of large partial quotients. We relate the value of J , to that ofSudler’s trigonometric product, and establish asymptotic upper and lowerbounds for such Sudler products in response to a question of Lubinsky. Quantum knot invariants arise in theoretical quantum physics, where a knot canbe regarded as the spacetime orbit of a charged particle. A typical example ofsuch an invariant is the n -colored Jones polynomial of the knot K = 4 (the figureeight knot), which is given by J ,n ( q ) = ∞ X N =0 q − nN N Y j =1 (1 − q n − j )(1 − q n + j ) , n ≥ , defined for roots of unity q . For a fixed q the mapping n J ,n ( q ) is periodic in n , and so the definition can be extrapolated backwards in n to give J , ( q ) = ∞ X N =0 | (1 − q )(1 − q ) · · · (1 − q N ) | (1)for a root of unity q . Note that both sums actually have only finitely manyterms for any root of unity q . The figure eight knot is the simplest hyperbolic Keywords: continued fractions, quadratic irrationals, quantum modular forms, coloredJones polynomials, Kashaev invariant, Sudler product.
Mathematics Subject Classification(2020): Throughout the paper empty sums equal 0, and empty products equal 1. K one obtains formulas for J K, which are of asomewhat similar but more complicated nature. The so-called Kashaev invariant h K i n = J K,n ( e (1 /n )), n = 1 , , . . . is another quantum invariant of the knot K ;here and for the rest of the paper e ( x ) = e πix . The Kashaev invariant plays akey role in the volume conjecture, an open problem in knot theory which relatesquantum invariants of knots with the hyperbolic geometry of knot complements.For more general background information, see [14]; in the context of our presentpaper we refer to [6] and the references therein.The functions J K, also have an interpretation as quantum modular forms asintroduced by Zagier [16], and are predicted by Zagier’s modularity conjecture tosatisfy an approximate modularity property. For the figure eight knot the modu-larity conjecture has been established [6, 9]; that is, the function J , ( q ) satisfies aremarkable modularity relation of the form J , ( e ( γr )) /J , ( e ( r )) ∼ ϕ γ ( r ), where γ ∈ SL ( Z ) acts on rational numbers r in a natural way, and the asymptoticsholds as r → ∞ along rational numbers with bounded denominators. The cocyclefunctions ϕ γ in general have jumps at every rational point, and consequently theasymptotics of J , ( e ( a/b )) for rational a/b is quite involved. It is known [2] that J , ( e (1 /n )) ∼ n / √ (cid:18) Vol(4 )2 π n (cid:19) as n → ∞ , where Vol(4 ) = 4 π Z / log (2 sin( πx )) d x ≈ . . Bettin and Drappeau [6, Theorem3] found the asymptotics of J , ( e ( a/b )) for more general rationals a/b in terms oftheir continued fraction expansions: if a/b = [ a ; a , . . . , a k ], a k >
1, is a sequenceof rational numbers such that ( a + · · · + a k ) /k → ∞ , thenlog J , ( e ( a/b )) ∼ Vol(4 )2 π ( a + · · · + a k ) . (2)The result applies to a large class of rationals, including 1 /n = [0; n ], as well as toalmost all reduced fractions with denominator n , as n → ∞ . Verifying a conjec-ture made by Bettin and Drappeau, in this paper we will show that (2) in generalfails to be true without the assumption ( a + · · · + a k ) /k → ∞ .The individual terms in (1) can be expressed in terms of the so-called Sudlerproducts, which are defined as P N ( α ) := N Y n =1 | πnα ) | , α ∈ R . (3)2his could also be written using q -Pochhammer symbols as P N ( α ) = | ( q ; q ) N | = | (1 − q )(1 − q ) · · · (1 − q N ) | with q = e ( α ) , but (3) seems to be the more common notation. The history of such productsgoes back at least to work of Erd˝os and Szekeres [8] and Sudler [15] around 1960,and they seem to arise in many different contexts (see [13] for references). Boundsfor such products also play a role in the solution of the Ten Martini Problem byAvila and Jitomirskaya [3]. It is somewhat surprising that, despite the obviousconnection between (1) and (3), we have not found a reference where both objectsappear together. A possible explanation is that (1) is only well-defined when q = e ( a/b ) with a/b being a rational, while the asymptotic order of (3) as N →∞ is only interesting when α is irrational. We will come back to this issue inProposition 3 below.Note that for any rational number a/b we have P N ( a/b ) = 0 whenever N ≥ b ;in particular this means that J , ( e ( a/b )) = P b − N =0 P N ( a/b ) . The asymptotics (2)a fortiori holds for more general functionals of the sequence ( P N ( a/b )) ≤ N b − X N =0 P N ( a/b ) c ! /c ∼ Vol(4 )4 π ( a + · · · + a k ) , (4)and also log max ≤ N
Let α be a quadratic irrational. For any real c > , log q k − X N =0 P N ( p k /q k ) c ! /c ∼ K c ( α ) k as k → ∞ nd log max ≤ N 1) = Q b − j =1 ( x − e ( j/b )). K c ( α ) − log λ ( α ) c ≤ K ∞ ( α ) ≤ K c ( α ) . (9)Finally, we establish an antisymmetry of the sequence ( P N ( a/b )) ≤ N
Proposition 2. For any rational number a/b with ( a, b ) = 1 , and any integer ≤ N < b , log P N ( a/b ) + log P b − N − ( a/b ) = log b. Note that Proposition 2 immediately implies thatlog max ≤ N
10 = [3; 6 , , , . . . ] satisfy K ∞ (cid:18) √ (cid:19) ≥ log 1 + √ ≈ . , K ∞ ( √ ≥ log(1 + √ ≈ . ,K ∞ (cid:18) √ (cid:19) ≥ log 3 + √ ≈ . , K ∞ ( √ ≥ log(2 + √ ≈ . ,K ∞ (cid:18) √ (cid:19) ≥ log 5 + √ ≈ . , K ∞ ( √ ≥ log(3 + √ ≈ . , (11)whereas Vol(4 )4 π a (cid:18) √ (cid:19) ≈ . , Vol(4 )4 π a ( √ ≈ . , Vol(4 )4 π a (cid:18) √ (cid:19) ≈ . , Vol(4 )4 π a ( √ ≈ . , Vol(4 )4 π a (cid:18) √ (cid:19) ≈ . , Vol(4 )4 π a ( √ ≈ . . 5n particular, the sequence of convergents of these quadratic irrationals violate (2),(4) and (5), demonstrating that the condition ( a + · · · + a k ) /k → ∞ cannot beremoved.For a quadratic irrational α with large partial quotients the constants K c ( α )and K ∞ ( α ) are nevertheless close to Vol(4 ) a ( α ) / (4 π ). Indeed, from results ofBettin and Drappeau [6, Theorem 2 and Lemma 15] it follows that for any rational a/b = [ a ; a , . . . , a k ], a k > J , ( e ( a/b )) = Vol(4 )2 π ( a + · · · + a k ) + O ( A + k log A ) (12)with A = 1 + max ≤ ℓ ≤ k a ℓ and a universal implied constant. A fortiori, for any real c > b − X N =0 P N ( a/b ) c ! /c = Vol(4 )4 π ( a + · · · + a k )+ O ( A + k max { , /c } log A ) (13)and log max ≤ N K c ( α ) ≥ (cid:18) c + 12 (cid:19) log λ ( α );in light of (7) and (9) this is nontrivial for 0 < c < 2. In particular, K ∞ ( α ) < K c ( α )for all small enough c , and the set { K c ( α ) : c > } is infinite for all quadraticirrationals.As we mentioned earlier, previous results on the Sudler product concerned theasymptotics of P N ( α ) as N → ∞ with a given irrational α , whereas J , ( e ( a/b )) = P b − N =0 P N ( a/b ) has been studied for rational a/b . To make these two types ofresults easier to compare, we prove the following simple “transfer principle”.6 roposition 3. Let α = [ a ; a , a , . . . ] be an irrational number with convergents p k /q k = [ a ; a , . . . , a k ] . For any integers k ≥ and ≤ N < q k , | log P N ( α ) − log P N ( p k /q k ) | ≪ log A k a k +1 with A k = 1 + max ≤ ℓ ≤ k a ℓ and a universal implied constant. In particular, for any real c > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log q k − X N =0 P N ( α ) c ! /c − log q k − X N =0 P N ( p k /q k ) c ! /c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ log A k a k +1 and (cid:12)(cid:12)(cid:12)(cid:12) log max ≤ N 0, thus disproving the conjecture of Erd˝os and Szek-eres. On the other hand, however, in the paper [11] by the same authors it wasshown that for the special irrational α = [0; a, a, a, . . . ] = ( √ a + 4 − a ) / P N ( α ) = 0 whenever a is sufficiently large. Thus, rather remarkably,the question whether lim inf P N ( α ) > α in a very sensitive way. Numerical experiments suggesteda similar change of behavior for large values of P N : for α = [0; a, a, a, . . . ] it wasconjectured that lim sup P N ( α ) /N < ∞ or = ∞ , depending on the size of a . Thisproblem was finally settled in [1], where it was proved that for α = [0; a, a, a, . . . ],lim inf N →∞ P N ( α ) > N →∞ P N ( α ) N < ∞ if a ≤ N →∞ P N ( α ) = 0 and lim sup N →∞ P N ( α ) N = ∞ if a ≥ . (19)Our trivial lower bound (7) exhibits the same delicate behavior. From (18)and a slightly stronger form of (19) it follows that for α = [0; a, a, a, . . . ] we have K ∞ ( α ) = log λ ( α ) if and only if a ≤ 5. In other words, in (11) we actuallyhave equality everywhere except for √ 10, in which case the inequality is strict. Itseems to be a difficult problem to give a complete characterization of all quadraticirrationals α such that K ∞ ( α ) = log λ ( α ), i.e. for which our trivial lower bound(7) is sharp. This is closely related to the problem of characterizing all quadraticirrationals α for which lim inf N →∞ P N ( α ) > P N ( α ) /N < ∞ , which isthe subject of an upcoming paper of Grepstad, Neum¨uller and Zafeiropoulos [12].The discussion above shows that K ∞ ( α ) ≥ K ∞ ( √ ) = log √ for allquadratic irrationals. We do not know whether K c ( α ) ≥ K c ( √ ) for all quadraticirrationals and all c > 0; in fact, we do not even know the precise value of K c ( √ ) for any c . We mention, however, that numerical evidence found byZagier [16] and Bettin and Drappeau [6] suggests that for the golden mean wehave log J , ( e ( p k /q k )) ≈ . k ; that is, K ( √ ) ≈ . α , Lubinsky [13] proved that | log P N ( α ) | ≪ log N ; equivalently, N − c ≪ P N ( α ) ≪ N c (20)with some constants c , c and implied constants depending on α . Let c ( α ) resp. c ( α ) denote the infimum of all c resp. c for which (20) holds; Lubinsky remarkedthat it is an interesting problem to determine these constants. The reflectionprinciple (17) immediately shows that given a badly approximable α and a realconstant c , we have P N ( α ) ≫ N − c ⇔ P N ( α ) ≪ N c , with implied constantsdepending on α . Therefore c ( α ) = c ( α ) + 1, which is a striking general relationthat seems not to have been noticed so far. Thus establishing the optimal valueof c and that of c in (20) are actually one and the same problem. Note thatthis relation also explains why the behavior of the liminf and the limsup in (18)and (19) changes at the same critical value of a . The results in our paper allowus to give a fairly precise answer to Lubinsky’s question in the case when α is aquadratic irrational; note that by (16) and (20), we have c ( α ) = K ∞ ( α ) / log λ ( α ).From (15) with c = ∞ we thus obtain that for any quadratic irrational α , c ( α ) = c ( α ) + 1 = Vol(4 )4 π · a ( α )log λ ( α ) + O (cid:18) log A ( α )log λ ( α ) (cid:19) with a universal implied constant. In particular, for α = [0; a, a, a, . . . ] we have c ( α ) = c ( α ) + 1 = Vol(4 )4 π · a log a + √ a +42 + O (1) . In this context, it is an interesting question to characterize those values of N forwhich particularly large resp. small values of P N ( α ) occur. It is also interesting toestimate the relative number of indices which generate such values of P N ( α ). Thiswould shed some light on the relation between the numbers K c ( α ) and K ∞ ( α ) inTheorem 1 and (16). Essentially, the problem is whether the sum P MN =0 P N ( α ) c isdominated by a very small number of indices N which produce particularly largevalues of P N , or if there are enough such indices so that the full sum is of a signif-icantly different asymptotic order than its maximal term. We plan to come backto all these questions in a future paper.Finally, we mention a further open problem. In [16] Zagier introduced thefunction h ( x ) = log ( J , ( e ( x )) /J , ( e (1 /x ))). A conjecture of Zagier, establishedby Bettin and Drappeau in [6], implies that h has jumps at all rational points.Zagier also suggested that h ( x ) is continuous at irrational values of x (more pre-cisely, since h is formally only defined for rational x , the conjecture is that h α be a quadratic irrational whose continued fraction expansion is of the simpleform α = [0; a, a, a, . . . ], and let p k /q k be its convergents. Then it is easily seenthat h ( p k /q k ) = log ( J , ( e ( p k /q k ))) − log ( J , ( e ( p k − /q k − ))). Thus, while wecannot prove that h ( p k /q k ) converges as k → ∞ , as a consequence of Theorem1 we can at least conclude that K − P Kk =1 h ( p k /q k ) converges as K → ∞ . Onecould call this Ces`aro-continuity along continued fraction convergents towards α .If h can indeed be continuously extended to α , then the only possible value is h ( α ) = 2 K ( α ) with K ( α ) from Theorem 1. So while our results can be seen asprogress towards Zagier’s problem, the continuity of h at irrational points remainsopen. From the discussion above, one might expect that the problem requiresdifferent approaches according to whether the partial quotients of α are large (say,as in the case ( a + · · · + a k ) /k → ∞ ), or small (say, bounded). The most difficultcase could be the one when the partial quotients of α are small, but there is noparticular structure such as periodicity. Recalling (8), to deduce (4) and (5) from (2) we only need to show that thecondition ( a + · · · + a k ) /k → ∞ implies that log b/ ( a + · · · + a k ) → 0; indeed,this will show that for any c > b − X N =0 P N ( a/b ) c ! /c ∼ log max ≤ N b − X N =0 P N ( a/b ) c ! /c = log max ≤ N
By the definition of Sudler products, for any integer0 ≤ N < b , P N ( a/b ) · b − Y n = N +1 | πna/b ) | = P b − ( a/b ) . (21)A simple reindexing shows that here b − Y n = N +1 | πna/b ) | = b − N − Y j =1 | π ( b − j ) a/b ) | = P b − N − ( a/b ) . As observed in (6), we also have P b − ( a/b ) = b . Hence (21) yieldslog P N ( a/b ) + log P b − N − ( a/b ) = log b, as claimed. Let α = [ a ; a , a , . . . ] be an arbitrary irrational number with convergents p k /q k =[ a ; a , . . . , a k ]. Let A k = 1 + max ≤ ℓ ≤ k a ℓ , and let k x k denote the distance froma real number x to the nearest integer. The sequence q k satisfies the recursion q k = a k q k − + q k − with initial conditions q = 1, q = a . Recall that for any k ≥ < n < q k , we have k nα k ≥ k q k − α k . Further, if k ≥ 1, or k = 0and a > 1, then 1 q k +1 + q k < k q k α k < q k +1 . (22)The main tool in the proof of the transfer principle is a bound on a cotangentsum proved by Lubinsky [13, Theorem 4.1], which states that for any k ≥ ≤ N < q k , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 cot( πnα ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (124 + 24 log A k ) q k . (23)The same bound holds in the rational setting as well, i.e. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 cot( πnp k /q k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (124 + 24 log A k ) q k . (24)Indeed, we can apply (23) to a sequence of irrational α ’s converging to p k /q k ,whose continued fraction expansions have initial segments identical to p k /q k =[ a ; a , . . . , a k ]. The same cotangent sum and various generalizations thereof in11he rational setting have been studied recently in [7], and used in [6] to establish(2). Cotangent sums have a long history in analytic number theory; some of themare known to satisfy interesting reciprocity formulas, and they also appear in theNyman–Beurling–B´aez-Duarte approach to the Riemann hypothesis. See [4, 5] formore details. Proof of Proposition 3. We consider the cases q k < 200 and q k ≥ 200 sepa-rately, starting with the former; the value 200 is of course basically accidental.First, assume that q k < k = 1 and a = 1, then N = 0 and we aredone; we may therefore assume that either k ≥ 2, or k = 1 and a > 1. For any0 < n < q k we thus have2 ≥ | πnα ) | ≥ k nα k ≥ k q k − α k ≥ q k + q k − > , and similarly 2 ≥ | πnp k /q k ) | ≥ k np k /q k k ≥ q k > . (25)Consequently, | log P N ( α ) | ≪ | log P N ( p k /q k ) | ≪ 1, and we are done provided a k +1 is bounded. For large a k +1 note that | sin( πnα ) − sin( πnp k /q k ) | ≤ πn | α − p k /q k | ≤ πa k +1 . In particular, by (25) we have (cid:12)(cid:12)(cid:12)(cid:12) sin( πnα )sin( πnp k /q k ) − (cid:12)(cid:12)(cid:12)(cid:12) < πa k +1 , and hence (cid:18) − πa k +1 (cid:19) ≤ P N ( α ) P N ( p k /q k ) ≤ (cid:18) πa k +1 (cid:19) . This finishes the proof in the case q k < q k ≥ q k ≥ q ℓ ≥ a ℓ for all 1 ≤ ℓ ≤ k ;in particular, q k ≥ A k − 1. From the assumption q k ≥ 200 we thus deduce q k ≥ p A k − ≥ √ A k . Using a trigonometric identity, we can write P N ( α ) P N ( p k /q k ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y n =1 sin( πnα )sin( πnp k /q k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y n =1 (1 + x n + y n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (26)where x n = cos( πn ( α − p k /q k )) − ,y n = sin( πn ( α − p k /q k )) cot( nπp k /q k ) . (cid:12)(cid:12)(cid:12)(cid:12) α − p k q k (cid:12)(cid:12)(cid:12)(cid:12) < q k q k +1 ≤ q k − q k q k − + 1 ! ≤ q k (cid:18) − A k + 1 (cid:19) . From the Taylor expansions of sine and cosine, for all 0 < n < q k , | x n | ≤ π n (cid:12)(cid:12)(cid:12)(cid:12) α − p k q k (cid:12)(cid:12)(cid:12)(cid:12) ≤ π q k ≤ A k , as well as | y n | ≤ | sin( πn ( α − p k /q k )) | sin( π/q k ) ≤ πn | α − p k /q k | π/q k − π / (6 q k ) ≤ − π / (6 q k ) (cid:18) − A k + 1 (cid:19) ≤ − π / (600 A k ) (cid:18) − A k + 1 (cid:19) ≤ − A k . The previous two estimates give | x n + y n | ≤ − / (4 A k ); the point is that x n + y n is bounded away from − | x | ≤ − / (4 A k ), e x − x log(4 A k ) ≤ x ≤ e x . Indeed, one readily verifies that the function e − x +2 x log(4 A k ) (1 + x ) attains its min-imum on the interval [ − / (4 A k ) , − / (4 A k )] at x = 0. Applying this estimatewith x = x n + y n in each factor of (26), we obtain P N ( α ) P N ( p k /q k ) = exp O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 ( x n + y n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + N X n =1 ( x n + y n ) log(4 A k ) !! . (27)Note that the right-hand side of (27) provides both an upper and a lower boundfor the quotient on the left-hand side. Since | x n | ≤ π n (cid:12)(cid:12)(cid:12)(cid:12) α − p k q k (cid:12)(cid:12)(cid:12)(cid:12) ≤ π a k +1 q k , the contribution of x n and x n in (27) is negligible: N X n =1 | x n | + N X n =1 x n log(4 A k ) ≪ a k +1 q k + log A k a k +1 q k . From Lubinsky’s bound on cotangent sums (24), summation by parts and | sin( π ( n + 1)( α − p k /q k )) − sin( πn ( α − p k /q k )) | ≤ π | α − p k /q k | ≪ a k +1 q k , 13e obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 sin( πn ( α − p k /q k )) cot( πnp k /q k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ log A k a k +1 . Finally, N X n =1 y n log(4 A k ) ≪ N X n =1 log A k a k +1 q k k np k /q k k ≤ q k − X j =1 log A k a k +1 q k k j/q k k ≪ log A k a k +1 , since the integers np k , 1 ≤ n ≤ N attain each nonzero residue class modulo q k atmost once. Hence (27) simplifies as P N ( α ) P N ( p k /q k ) = exp (cid:18) O (cid:18) log A k a k +1 (cid:19)(cid:19) , which proves the proposition. Fix a quadratic irrational α . Throughout this section constants and impliedconstants depend only on α . The continued fraction expansion is of the form α = [ a ; a , . . . , a s , a s +1 , . . . , a s + p ], where the overline means period. As be-fore, p k /q k = [ a ; a , . . . , a k ] denotes the k -th convergent to α ; further, let δ k = ( − k ( q k α − p k ). The sequences q k and p k satisfy the same recursion; conse-quently, δ k = − a k δ k − + δ k − for all k ≥ 2. If k ≥ 1, or k = 0 and a > 1, then δ k = k q k α k .For any 1 ≤ r ≤ p let T r = (cid:18) a s + r + p (cid:19) · · · (cid:18) a s + r +1 (cid:19) . The recursion q k = a k q k − + q k − can be written in the form T mr (cid:18) q s + r − q s + r (cid:19) = (cid:18) q s + mp + r − q s + mp + r (cid:19) , m = 1 , , . . . Observe that det T r = ( − p , and that tr T r does not depend on r . Thereforethe eigenvalues η and µ of T r are the same for all 1 ≤ r ≤ p . Since q k → ∞ exponentially fast, we have, say, η > µ = ( − p /η . Consequently, therecursions for q k and δ k have solutions q s + mp + r = C r η m + D r ( − mp η − m ,δ s + mp + r = E r η − m (28)14ith some constants C r , E r > D r , 1 ≤ r ≤ p . In particular, log q k ∼ (log λ ( α )) k with λ ( α ) = η /p . Lemma 1. For any k ≥ we have κδ k ≤ δ k +1 ≤ (1 − κ ) δ k with some constant κ > . Proof. We claim that δ k ≤ ( a k +2 + 2) δ k +1 for all k ≥ 0. Indeed, if k = 0, a = 1this can be verified “by hand”; else, from (22) we obtain δ k < q k +1 ≤ a k +2 + 2 q k +2 + q k +1 < ( a k +2 + 2) δ k +1 . On the other hand, δ k +1 ≤ a k +2 δ k +1 = δ k − δ k +2 ≤ δ k − a k +3 + 2 δ k +1 , and hence δ k +1 ≤ δ k ( a k +3 + 2) / ( a k +3 + 3). The claim thus follows with κ =1 / (max k ≥ a k + 3). The fundamental object in the proof of Theorem 1 are “perturbed” versions of theSudler product defined as P q k ( α, x ) := q k Y n =1 | π ( nα + ( − k x/q k )) | , x ∈ R . Perturbed Sudler products were first introduced by Grepstad, Kaltenb¨ock andNeum¨uller [10], and have since been used in [1] and [12]. The relevance of thesefunctions come from the Ostrowski expansion of integers, which we now recall.Any integer N ≥ N = P ∞ k =0 b k q k ,where 0 ≤ b ≤ a − ≤ b k ≤ a k +1 , k ≥ b k − = 0 whenever b k = a k +1 . Of course, the series only has finitely manynonzero terms; more precisely, if 0 ≤ N < q k +1 , then b k +1 = b k +2 = · · · = 0. Givenan integer N ≥ N = P ∞ k =0 b k q k , let us introduce thenotation ε k ( N ) := q k ∞ X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ , where δ ℓ = ( − ℓ ( q ℓ α − p ℓ ) was already defined at the beginning of this section.Further, we shall write f ( x ) = | πx ) | .15 emma 2. For any integer N ≥ with Ostrowski expansion N = P ∞ k =0 b k q k , P N ( α ) = ∞ Y k =0 b k − Y b =0 P q k ( α, bq k δ k + ε k ( N )) . Proof. Note that only finitely many factors are different from 1. Let N k = P ∞ ℓ = k b ℓ q ℓ . Then N = N ≥ N ≥ N ≥ · · · , and N k = 0 for all large enough k . By the definition of Sudler products, P N ( α ) = ∞ Y k =0 N k Y n = N k +1 +1 f ( nα )= ∞ Y k =0 b k q k Y n =1 f ( nα + N k +1 α )= ∞ Y k =0 b k − Y b =0 q k Y n =1 f nα + bq k α + ∞ X ℓ = k +1 b ℓ q ℓ α ! = ∞ Y k =0 b k − Y b =0 q k Y n =1 f nα + ( − k bδ k + ∞ X ℓ = k +1 ( − ℓ b ℓ δ ℓ ! = ∞ Y k =0 b k − Y b =0 P q k ( α, bq k δ k + ε k ( N )) . The main message of the next lemma is that P q k ( α, x ) has a positive lowerbound at all points which appear in the claim of Lemma 2. From now on let1 ≤ [ k ] ≤ p denote the remainder of k − s modulo p , where s is the length of thepre-period in the continued fraction for α ; that is, if k = s + mp + r , then [ k ] = r . Lemma 3. (i) For any integer N ≥ with Ostrowski expansion N = P ∞ k =0 b k q k , any k ≥ and any ≤ b < b k we have P q k ( α, bq k δ k + ε k ( N )) ≫ uniformly in N , k and b .(ii) There exist compact intervals I r , ≤ r ≤ p , and a constant k > s with thefollowing properties. First, for any integer N ≥ with Ostrowski expansion N = P ∞ k =0 b k q k , any k ≥ k and any ≤ b < b k we have bq k δ k + ε k ( N ) ∈ I [ k ] .Second, P q k ( α, x ) ≫ on I [ k ] uniformly in k ≥ k . roof. Fix an integer N ≥ P ∞ k =0 b k q k , and integers k ≥ ≤ b < b k . We necessarily have b k > 0; in particular, k ≥ 1, or k = 0and a > 1. Observe that ε k ( N ) = q k ∞ X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ ≤ q k ( a k +3 δ k +2 + a k +5 δ k +4 + · · · )= q k (( δ k +1 − δ k +3 ) + ( δ k +3 − δ k +5 ) + · · · )= q k δ k +1 . Since b k > 0, by the extra rule of the Ostrowski expansions we have b k +1 < a k +2 .Therefore ε k ( N ) = q k ∞ X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ ≥ − q k (( a k +2 − δ k +1 + a k +4 δ k +3 + · · · )= − q k ( − δ k +1 + ( δ k − δ k +2 ) + ( δ k +2 − δ k +4 ) + · · · )= − q k ( δ k − δ k +1 ) . Letting κ > | ε k ( N ) | ≤ (1 − κ ) q k δ k .Consider now P q k ( α, bq k δ k + ε k ( N )) = q k Y n =1 f (( n + bq k ) α + ( − k ε k ( N ) /q k ) . For each 1 ≤ n ≤ q k we have n + bq k ≤ a k +1 q k < q k +1 , and hence by the bestapproximation property of continued fractions (cid:13)(cid:13) ( n + bq k ) α + ( − k ε k ( N ) /q k (cid:13)(cid:13) ≥ k q k α k − | ε k ( N ) | /q k ≥ κδ k . Consequently, P q k ( α, bq k δ k + ε k ( N )) ≫ fixed k ≥ 0. It will thus be enoughto prove Lemma 3 (ii), and Lemma 3 (i) will follow.We now prove Lemma 3 (ii). Observe that (28) implies q s + mp + r δ s + mp + r → B r as m → ∞ , where B r = C r E r > 0, 1 ≤ r ≤ p , are constants. Define I r = [ − (1 − κ/ B r , ( a s + r +1 − κ/ B r ] (1 ≤ r ≤ p ) . These intervals, together with some constant k , to be chosen, satisfy the claim.Choosing k large enough, for all k ≥ k and all 0 ≤ b < a k +1 we have bq k δ k +[ − (1 − κ ) q k δ k , (1 − κ ) q k δ k ] ⊆ I [ k ] . In particular, bq k δ k + ε k ( N ) ∈ I [ k ] for all N ≥ k ≥ k and all 0 ≤ b < b k .Now let k ≥ k and x ∈ I [ k ] be arbitrary, and let us prove a lower bound for P q k ( α, x ). Then x = bB [ k ] + y for some appropriate integer 0 ≤ b < a k +1 , and some | y | ≤ (1 − κ/ B [ k ] . Let z = ( − k ( y + b ( B [ k ] − q k δ k )) q k , | z | ≤ (1 − κ/ B [ k ] + ( a k +1 − | q k δ k − B [ k ] | q k ≤ (1 − κ/ δ k (29)provided k was chosen large enough. With this choice of z we have f ( nα + ( − k x/q k ) = f (( n + bq k ) α + z ) , and so P q k ( α, x ) Q ( b +1) q k n = bq k +1 f ( nα ) = Q q k n =1 f (( n + bq k ) α + z ) Q q k n =1 f (( n + bq k ) α )= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q k Y n =1 (cos( πz ) + sin( πz ) cot( π ( n + bq k ) α )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (30)where the last equation follows from standard trigonometric identities. Using k ( n + bq k ) α k ≥ k q k α k = δ k and (29), we obtain | cos( πz ) − πz ) cot( π ( n + bq k ) α ) | ≤ | cos( πz ) − | + | sin( πz ) | sin( πδ k ) ≤ π − κ/ δ k + π (1 − κ/ δ k πδ k − π δ k / ≤ − κ/ k was chosen large enough; the point is that each factor in (30) isbounded away from 0. Following the steps in the proof of Proposition 3 (in par-ticular, recalling the cotangent sum estimate (23)), we thus deduce that P q k ( α, x ) Q ( b +1) q k n = bq k +1 f ( nα )= exp O δ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q k X n =1 cot(( n + bq k ) α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + δ k q k X n =1 cot (( n + bq k ) α ) !! = exp ( O (1)) . On the other hand, a general result of Lubinsky [13, Proposition 5.1] implies that1 ≪ P N ( α ) ≪ N contains ≪ ( b +1) q k Y n = bq k +1 f ( nα ) = P ( b +1) q k ( α ) P bq k ( α ) ≫ , and hence P q k ( α, x ) ≫ 1, as claimed. 18 .2 The limit functions The perturbed Sudler products P q k ( α, x ) were shown to converge to an explicitlygiven limit function for the special irrationals α = [0; a, a, a, . . . ] in [1]. This waslater generalized to all quadratic irrationals by Grepstad, Technau and Zafeiropou-los [12], whose result we now quote. Note that the periodicity of the continuedfraction expansion is crucial for such a limit relation; that is, there is no analogueof the following theorem which holds for all badly approximable α . Lemma 4 ([12]) . For any quadratic irrational α there exist continuous functions G r ( α, x ) , ≤ r ≤ p such that P q s + mp + r ( α, x ) → G r ( α, x ) pointwise on R as m →∞ . Moreover, the convergence is uniform on any compact interval, and G r iscontinuously differentiable on any open interval on which G r > . Lemma 3 implies that G r > 0, and consequently that log G r is Lipschitz on thecompact interval I r . The main idea of the proof of Theorem 1 is to replace theperturbed Sudler product P q k ( α, bq k δ k + ε k ( N )) by its limit G [ k ] ( α, bq k δ k + ε k ( N ))in the claim of Lemma 2. To this end, for any integer N ≥ N = P ∞ k =0 b k q k let G N ( α ) = ∞ Y k = k b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N )) . Lemma 5. For any real c > we have log X ≤ N By Lemma 2, for all 0 ≤ N < q ℓ with Ostrowski expansion N = P ℓ − k =0 b k q k , P N ( α ) G N ( α ) = k − Y k =0 P q k ( α, bq k δ k + ε k ( N )) · ℓ − Y k = k b k − Y b =0 P q k ( α, bq k δ k + ε k ( N )) G [ k ] ( α, bq k δ k + ε k ( N )) . Let ε > ≤ k ≤ k − e O (1) . Using Lemma 3 (ii) and the fact that by Lemma 4 we have P q s + mp + r ( α, x ) → G r ( α, x ) uniformly on I r , each factor with k ≤ k < K ( ε ) is e O (1) , and each factorwith K ( ε ) ≤ k ≤ ℓ − e O ( ε ) with some constant K ( ε ). Hence P N ( α ) G N ( α ) = e O (1+ K ( ε )+ εℓ ) . 19n particular,log X ≤ N For any real c > , the sequences c m := log X ≤ N Let 0 ≤ N < q k +( m + n ) p be an integer with Ostrowski expansion N = P k +( m + n ) p − k =0 b k q k . Consider the natural factorization G N ( α ) = k + mp − Y k = k b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N )) k +( m + n ) p − Y k = k + mp b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N )) . (31)Let us write N = N + N with N = P k + mp − k =0 b k q k and N = P k +( m + n ) p − k = k + mp b k q k .The plan of the proof is to show that, making a small error, we can replace ε k ( N )in (31) by ε k ( N ) and ε k ( N ), respectively, so that G N ≈ G N G N . Then we willshow that we can replace N by a number N ∗ having the “shifted” Ostrowskirepresentation N ∗ = P k + np − k = k b k + mp q k , and obtain G N ≈ G N G N ∗ . This approxi-mate shift-invariance is crucial for the argument, and comes from the periodicityof the continued fraction expansion of α . From G N ≈ G N G N ∗ we can deduce that c m + n ≈ c m + c n and c ∗ m + n ≈ c ∗ m + c ∗ n , which is what we want to prove.20ow we make this precise. Note that for any k ≤ k ≤ k + mp − | ε k ( N ) − ε k ( N ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q k k +( m + n ) p − X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ − q k k + mp − X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ q k k +( m + n ) p − X ℓ = k + mp δ ℓ ≪ q k δ k + mp . Since log G r is Lipschitz on I r , the previous estimate implies that the first factorin (31) is k + mp − Y k = k b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N ))= k + mp − Y k = k b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N )) e O ( q k δ k mp ) = e O ( q k mp δ k mp ) k + mp − Y k = k b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N ))= e O (1) G N ( α ) . (32)Now let N ∗ = P k + np − k = k b k + mp q k ; observe that this is a valid Ostrowski expan-sion of 0 ≤ N ∗ < q k + np . Using (28), for any k + mp ≤ k ≤ k + ( m + n ) p − ≤ b < b k , | bq k δ k − bq k − mp δ k − mp | ≪ η m − k/p . Similarly, | ε k ( N ) − ε k − mp ( N ∗ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q k k +( m + n ) p − X ℓ = k +1 ( − k + ℓ b ℓ δ ℓ − q k − mp k + np − X ℓ = k − mp +1 ( − k − mp + ℓ b ℓ + mp δ ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k +( m + n ) p − X ℓ = k +1 ( − k + ℓ b ℓ ( q k δ ℓ − q k − mp δ ℓ − mp ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ k +( m + n ) p − X ℓ = k +1 η m − k/p − ℓ/p ≪ η m − k/p . k +( m + n ) p − Y k = k + mp b k − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N ))= k +( m + n ) p − Y k = k + mp b k − Y b =0 G [ k ] ( α, bq k − mp δ k − mp + ε k − mp ( N ∗ )) e O ( η m − k/p ) = e O (1) k + np − Y k = k b k + mp − Y b =0 G [ k ] ( α, bq k δ k + ε k ( N ∗ ))= e O (1) G N ∗ ( α ) . (33)From (31), (32) and (33) we finally obtain the approximate factorization G N ( α ) = G N ( α ) G N ∗ ( α ) e O (1) . As N runs in the interval 0 ≤ N < q k +( m + n ) p ,we obtain each pair ( N , N ∗ ) ∈ [0 , q k + mp ) × [0 , q k + np ) at most once by the unique-ness of Ostrowski expansions. Therefore X ≤ N According to Lemma 6, the sequence c m + K is subadditivewith a large enough constant K . Since c m ≥ log G ( α ) = 0, Fekete’s subadditivelemma shows that the sequence c m m = 1 m log X ≤ N References [1] C. Aistleitner, N. Technau and A. Zafeiropoulos: On the order of magnitude of Sudlerproducts. Preprint, available at arXiv:2002.06602.[2] J. Andersen and S. Hansen: Asymptotics of the quantum invariants for surgeries on thefigure 8 knot. J. Knot Theory Ramifications 15 (2006), 479–548.[3] A. Avila and S. Jitomirskaya: The Ten Martini Problem. Ann. of Math. 170 (2009), 303–342.[4] S. Bettin and B. Conrey: Period functions and cotangent sums. Algebra Number Theory7 (2013), 215–242.[5] S. Bettin, and J. B. Conrey: A reciprocity formula for a cotangent sum. Int. Math. Res.Not. IMRN 24 (2013), 5709–5726.[6] S. Bettin and S. Drappeau: Modularity and value distribution of quantum invariants ofhyperbolic knots. Preprint, available at arXiv:1905.02045.[7] S. Bettin and S. Drappeau: Partial sums of the cotangent function. Preprint, available atarXiv:1905.01954.[8] P. Erd˝os and G. Szekeres: On the product Q nk =1 (1 − z a k ) . Acad. Serbe Sci. Publ. Inst.Math. 13 (1959), 29–34.[9] S. Garoufalidis and D. Zagier: Quantum modularity of the Kashaev invariant. In prepara-tion.[10] S. Grepstad, L. Kaltenb¨ock and M. Neum¨uller: A positive lower bound for lim inf N →∞ Q Nr =1 | sin πrφ | . Proc. Amer. Math. Soc. 147 (2019), 4863–4876.[11] S. Grepstad, L. Kaltenb¨ock and M. Neum¨uller: On the asymptotic behaviour of the sineproduct Q nr =1 | sin πrα | . Preprint, available at arXiv:1909.00980.[12] S. Grepstad, M. Neum¨uller and A. Zafeiropoulos: On the order of magnitude of Sudlerproducts II. Preprint.[13] D. Lubinsky: The size of ( q ; q ) n for q on the unit circle. J. Number Theory 76 (1999),217–247.[14] H. Murakami and Y. Yokota: Volume conjecture for knots. Springer Briefs in MathematicalPhysics, 30. Springer, Singapore, 2018. 15] C. Sudler Jr.: An estimate for a restricted partition function. Quart. J. Math. Oxford Ser.15 (1964), 1–10.[16] D. Zagier: Quantum modular forms. . In particular, we havelog J , ( p k /q k ) ∼ K ( α ) k as k → ∞ . The proof of Theorem 1 is based on the self-similar structure of quadratic irra-tionals; that is, on the periodicity of the continued fraction. In fact, it is notdifficult to construct a badly approximable α for which the result is not true.Quadratic irrationals exhibit a remarkable deviation from the universal behav-ior of irrationals with unbounded partial quotients. In contrast to (2), (4) and (5),the constants K c ( α ) and K ∞ ( α ) in Theorem 1 are in general not equal to eachother, or to Vol(4 ) a ( α ) / (4 π ); here a ( α ) = lim k →∞ ( a + · · · + a k ) /k denotes theaverage partial quotient, which is of course simply the average over a period inthe continued fraction expansion. We have not been able to calculate the precisevalue of K c ( α ) for any specific quadratic irrational and for any c ; as far as we cansay, this seems to be a difficult problem. The precise value of K ∞ ( α ) is known forsome quadratic irrationals with very small partial quotients, as a consequence ofresults in [1]; for α with larger partial quotients, calculating K ∞ ( α ) precisely alsoseems to be difficult.However, it is possible to give fairly good general upper and lower bounds for K c ( α ) and K ∞ ( α ) in terms of the partial quotients. Recall that for any quadraticirrational α we have log q k ∼ (log λ ( α )) k as k → ∞ with some constant λ ( α ) > λ ( α ) from the continued fraction.We start with three simple observations. First, for any rational number a/b with( a, b ) = 1, the identity P b − ( a/b ) = b − Y n =1 | − e ( na/b ) | = b − Y j =1 | − e ( j/b ) | = b (6)provides the trivial lower bound max ≤ N b b − X N =0 P N ( a/b ) c ! /c ≤ max ≤ N
q k ∼ (log λ ( α )) k , as before. For an arbitraryirrational α the reflection principle becomes, with the notation of Proposition 3,log P N ( α ) + log P q k − N − ( α ) = log q k + O (cid:18) log A k a k +1 (cid:19) ;in particular,log max ≤ N
α . Note that the relations in the previous line also follow fromthe identity (6) and the transfer and reflection principles; in fact, the limsuprelation is a far reaching generalization of our trivial lower bound (7). More re-cently Grepstad, Kaltenb¨ock and Neum¨uller [10] established the remarkable re-lation lim inf P N ( √ ) >