UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE

Transcription

1 uniform distribution theory DOI: 055/udt Unif Distrib Theory 2 207), no2, 24 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE Christopher J White ABSTRACT We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, providedα is badly-approximable We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α Communicated by Michael Drmota Introduction Some notation Let α =[a 0 ; a,] denote the continued fraction expansion of α R \ Q We write x for the distance from x to the nearest integer The convergents p n /q n =[a 0 ; a,,a n ], where p n,q n ) =, give good approximations to α We call {q n } n N the sequence of denominators of α We say that an irrational number α is badly-approximable if there exists ε α > 0 such that for all p, q Z, p, q) =,wehave α p q > ε α q 2 These correspond precisely with those numbers α for which there exists N N such that, a n α) N for all n N The set of all badly-approximable numbers is a set of Lebesgue measure zero 200 M a t h e m a t i c s Subject Classification: J70, L03, L07 K e y w o r d s: continued fraction, badly-approximable α, double-exponential sum, discrepancy, Koksma-Hlawka inequality, Ostrowski expansion The author s research is supported by an EPSRC Doctoral Training Grant

2 CHRISTOPHER J WHITE When α is badly approximable, we have the helpful bound that q n α > ε α q n Since convergents give the best approximations for the distance to the nearest integer see [7]), this means that for m q n+ we have the bound mα > ε α q n We write fn) = Ogn)) to mean that there exists a constant C, whichdoes not depend on n), such that fn) C gn) for all n N Finally, we define the discrepancy of a sequence Definition Let x n ) be a sequence of real numbers For N N the discrepancy of x n ) modulo one, D N x n ), is defined as: N D N {x m }):= sup χ I x m ) N I, I R/Z m= where I denotes an interval and χ I is the characteristic function of I 2 Double exponential sums In [9] S i n a i and U l c g r a i studied double trigonometric sums of the form T M α) = M M M m=0 n=0 enmα) 2) We want to determine when the absolute value of this sum is bounded uniformly ie, by a constant which depends only on α) over some subsequence M A N This will, obviously, depend on the Diophantine properties of α and the subsequence A We will see that the problem of bounding this sum depends importantly on controlling sums such as M M {{mα}} 22) m= 2

3 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE Here {{x}} := { {x}, x [0, 2 ], {x}, x 2, 0), is the signed fractional part of x R In [9] the following is proved Theorem 2 Sinai, Ulcigrai) Let α be badly-approximable Consider the following double trigonometric sum: T M α) := M M M m=0 n=0 enmα) Then there exists a constant C = Cα) > 0 such that T M C α for all M {q n } n N Remark 3 The sum here is an example of a 2-dimensional finite theta sum In [2] C o s e n t i n o and F l a m i n i o prove bounds for far more general g- -dimensional finite theta sums A special case of one of their results implies that the above theorem is true for all M N Our main theorem generalises Theorem 2 Theorem 4 Let α R \ Q Then there exists a constant C = Cα) > 0 such that T qn C α max { } log2 maxi n {a i }), a n+ for all n N 23) Remark 5 By examining signs it appears that the upper bound here is close to best possible Equation 3) in [] gives a lower bound for the largest terms in a sum that we will consider While it is true that we use the triangle inequality earlier in our calculation, it does not make our estimate so much larger 3

4 2 Reducing T M CHRISTOPHER J WHITE 2 Proof of main result Following the methods in [9], we split T M into two separate sums By summing the terms for n =0,,M, we can rewrite 2) as M T M =+ M m= Then we can write T M =+S M S M,where M emmα) emα) emmα) emα) S M := M m= and S M := M M m= We will prove that there exist constants C,C R such that { } log2 S q n C maxi n {a i }) max, a n+ and S q n C for all n N These constants will depend only on α 2) emα) 22) 23) 22 The sum S M 22) Let us consider the less intimidating sum first We want to show that there exists a C R such that S q n C for all n N Note that in [3], H a r d y and L i t t l e w o o d prove a similar theorem Theorem 2 Hardy, Littlewood) Let α be badly-approximable Then there exists C > 0 such that S M C for each M N + We proceed by calculating real and imaginary parts emα) = 2 i 2 cotπmα) The Taylor series expansion of cot x is ) n 4 n B 2n cot x = x 2n = 2n)! x x 3 + x3 45 n=0 with radius of convergence 0 < x <πhereb n is the nth Bernoulli number 4

5 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE Note that due to the symmetry of cot x, cotπmα) =cotπ{{mα}}) So we can write ) ) n 4 n B 2n ) 2n cotπmα) = + π{{mα}} π{{mα}} 2n)! n= Now the series on the right is negative and it takes values strictly between 0 when {{mα}} is close to 0) and when{{mα}} is close to ± 2 ) Hence, in order to prove that S q n is bounded by a uniform constant for all n N, wehavetoprovethefollowing: Lemma 22 Let α R Then there exists C = Cα) > 0 such that, q n C for all n N 22) q n {{mα}} m= We will consider two different proofs of Lemma 22 The first one is simpler, while the latter one will be applicable to estimating S M as well The second proof is also malleable to proving Theorem 2 23 Koksma-Hlawka Proof of Lemma 22 Recall the Koksma-Hlawka inequality Lemma 23 Let f be a real function with period of bounded variation Then for every sequence {x m } and every integer N, N fx m ) fx)dx N V f)d Nx m ), N m= where V f) is the total variation of the function 0 We wish to apply this inequality with fx) = {{x}}, x m = {mα}, and N = q n Therefore we have to restrict the domain on which we define our function, in order to ensure that it is integrable We are able to use the following from [8] α p n > 23) 2q n q n q n 5

6 So, for all m N = q n, we have CHRISTOPHER J WHITE mα > 2q n Hencewecanrestrictthedomainoff to the interval [ 2q n, 2q n ] Furthermore, since f is anti-symmetric about /2, the integral of f over this interval is equal to 0 The total variation, V f), of f is n p sup P {{x i+ }} {{x i }}, i= where P is a partition of [ 2q n, 2q n ] As f is monotone in this interval, the total variation is maximised when we take the trivial partition that is the two endpoints) Therefore V f) =4q n Finally, we move on to considering the Discrepancy Lemma 56 from [5] states that where D N mα) 3 N = r t j, j=0 r q j t j j=0 is the Ostrowski expansion of N This is defined in the next subsection see Definition 25) but all we need to know here is that if N = q n,thent n =and t i = 0 for all i n SoD qn mα) 3 Finally, we can apply all the estimates we have with N = q n andf & {x m } as above) q n fx m ) q n ) fx)dx D q n x m )V f) m= 0 D qn x m )+ ) V f) Hence, q n q n m= {{mα}} 6 4 4q n =6q n Here we used the obvious fact that D M x m ) D M+ x m )+ 6

7 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE 24 The sum S M 2) We move on to considering the sum S M := M M m= emmα) emα) We will write this sum as a telescoping series and then take advantage of some cancellation to reduce our situation to considering the sum S M 22) Firstly, M m= M emmα) emα) = m= emmα) e Mm +)α )) m k= ekα) 24) M + em 2 α) ekα) 242) k= We then consider the outer part of the sum on the right hand side of 24) for M = q n ), emq n α) e m +)q n α ) = emq n α) e m +)q n α ) = emq n α) emq n α)eq n α) = eq n α) ) emq n α) Inabsolutevaluethisislessthan2π/q n+ Now using the triangle inequality and Lemma 22, we see that 23) results from the following lemma Lemma 24 For all α R \ Q and for all m q n, m ) {{kα}} = O q n max {, log a i} i n k= To prove this Lemma we will need to introduce some different techniques, which will also yield a new proof of Lemma The Ostrowski proof of Lemmas 22 and 24 Our alternative proof of Lemma 22 will involve decomposing the sum in 22) into segments where there is some obvious cancellation Definition 25 Let α be irrational Then for every n N there exists a unique integer M 0 and a unique sequence {c k+ } k=0 of integers such that q M m<q M+ and m = c k+ q k, k=0 7

8 CHRISTOPHER J WHITE with 0 c < a, 0 c k+ a k+ for k, c k = 0 whenever c k+ = a k+ for some k, and c k+ = 0 for k>m This is known as the Ostrowski expansion We will consider segments of our sum which spread out in the unit interval We take our inspiration from a set of intervals discussed in [6] Definition 26 Special intervals) For fixed α define Ai) to be the collection of non-negative integers n with Ostrowski expansions of the form n = c k+ q k k=i Then for each i N and for each γ R/Z we define a subset J i, γ) which turns out to be an interval, see [6]) of R/Z by J i, γ) =γ + {nα : n Ai)} These intervals have some very nice properties such as N sup sup χ N N J R/Z J nα) N J K, n= where K is a universal constant and the inner supremum is taken over all special intervals J for α We will use what these intervals tell us about the distribution of nα on the unit interval to achieve cancellation in 22) and Let n m = c i+ q i q n, 0 c i+ a i+ i=0 i ni, c) := c j+ q j + cq i j=0 We will use this decomposition to sum up to m m k= n {{kα}} = i=0 c i+ ni,c)+q i l=ni,c)+ {{lα}} Note that ni, c)+q i = ni, c +) and ni, c i+ ) + q i = ni +, 0) 8

9 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE Let us consider a situation, where we are studying ni,c)+q i l=ni,c)+ {{lα}} 25) We wish to approximate α by p i /q i and achieve almost) complete cancellation in the main term that we get Obviously, problems can occur Specifically, if l p i 0q i ), then we do not want to divide by 0, so we want to isolate these terms and deal with them separately Note that since p i,q i ) =, we have a complete set of residue classes modulo q i, so in each sum 25) we will have exactly one term, l =c +)q i, where this happens Also, there exists r q i such that ni, 0) + r = q i, ni, ) + r =2q i, ni, c i+ ) + r = c i+ q i So we can consider all of these terms separately Finally, we consider summing over a complete set of residue classes modulo q i We will first consider the simple case, k q i ), which will give us a second proof of Lemma 22 We write Now α = p i + ξ i, q i q i q i+ q i k= qi {{kα}} = Now we use the fact that {{ kpi + kξ }} i = q i q i q i+ k= where 2 < ξ i < {{k p i q i + kξ i q i q i+ }} 252) {{ kpi q i }} {{ }} kξi +, q i q i+ unless perhaps if kp i q i 2 modulo q i when 2 q i ), or if kp i q i± 2 when 2 q i +) Now 252) equals q i k= {{k p i q i }} ) +{{ kp i q i }} kξ i + O) q i q i+ The one or two extra term/s mentioned just above have been removed from the sum and are accounted for by the O) term 9

10 Furthermore, +{{ kp i q i }} kξ i q i q i+ CHRISTOPHER J WHITE ) {{ }} kpi kξ i = q i q i q i+ {{ }} 2 ) 2 kpi kξi + q i q i q i+ There exists n k such that n k q i andn k kp i n k as follows { n nk, n k q i k := 2, n k q i, n k > q i 2 Then, mod q i Now we define {{ }} kpi kξ i = kξ i q i q i q i+ n k q i+ We then know that for all k, {{ }} {{ }} 2 ) 2 kpi kξ i kpi kξi kξ i + = C k q i q i q i+ q i q i q i+ n k q i+ We need n k 2 in order to have a uniform bound over k for the constant C k When this is the case 2 <C k < 2 apart from the one or two exceptions mentioned previously) So we have to isolate another two terms We write k, k for the numbers, where k p i mod q i and k p i mod q i, respectively So 252) becomes q i 2 n k =2 ) ) {{ n k ql }} + C kξi q i k + n k )2 q i+ {{k α}} + {{k α}} + O) = q i 2 n k =2 C k kξi q i n k )2 q i+ ) + Oq i ) Here we used the basic approximation from K h i n c h i n 23) to deal with the two extra terms) 20

11 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE By the rearrangement inequality, see [4], Theorem 368), this first sum is smaller in modulus) than 4q i ) 3 2 +, which in turn is bounded above by 4q i So q i k= l=ni,c)+ {{kα}} = Oq i), as required Now, we move on to a proof of Lemma 24 We wish to prove, for all i), that c i+ ni,c)+q i {{lα}} = Oq i+ log c i+ ) Note that if we sum ni,c)+q i l=ni,c)+ {{lα}}, then a similar argument to the proof of Lemma 22 shows that this is equal to Oq i )+ {{k,c) α}} + {{k,c) α}} + {{c +)q i α}}, where ni, c)+ k ±,c) ni, c)+q i, and k ±,c) p i ± mod q i Clearly, k ±,c) = k ±,0) + cq i Furthermore, as ni, 0) <q i, k ±,ci+ r) <n i, c i+ r) ) + q i <q i+ r )q i Now, we calculated earlier that Letting k = k,0), {{kα}} = {{k p i q i }} ) +{{ kp i q i }} kξ i q i q i+ {{k,0) α}} = q i + k,0)ξ i q i+ = ) q i q i+ q i+ + k,0) ξ i 2

12 CHRISTOPHER J WHITE Hence, and also {{k,c) α}} = q i q i+ q i+ +k,0) + cq i )ξ i {{k,c) α}} = q i q i+ q i+ k,0) + cq i )ξ i Now, without loss of generality, assume that ξ i > 0 Then Hence, c i+ {{k,c) α}} <q i for all c ni,c)+q i l=ni,c)+ c i+ {{lα}} = Oq i+) + {{c +)q i α}} c i+ + q i q i+ q i+ k,0) + cq i )ξ i = Oq i+ )+Oq i+ log c i+ ) c i+ q i q i+ + q i+ k,0) + cq i )ξ i Finally, using k,0) q i + q i and ξ<), c i+ q i q i+ q i+ k,0) + cq i )ξ i q i+ c i+ + + q i+ 2 + q i+ +2q i+ = Oq i+ log c i+ ) As this is true for all i, the condition for Lemma 24 follows Remark 27 Equation 3) in [] tells us that the sum c i+ can be no smaller than Oq i+ log c i+ ) {{c +)q i α}} 22

13 UPPER BOUNDS FOR DOUBLE EXPONENTIAL SUMS ALONG A SUBSEQUENCE Remark 28 In our final calculation we have ignored the cancellation between the positive and negative terms However, when c i+ a i+ /2, for example, we get very little cancellation and our main term is Oq i+ log a i+ ) Remark 29 In fact since convergents p i /q i give lower bounds for α when i is even this implies that ξ i is positive when i is even and negative when i is odd) Now from the well-known formula for continued fractions p i q i p i q i = ) i, we see that for the negative terms we separated, k,c) = cq i + q i These correspond to the semiconvergents cp i + p i cq i + q i in the continued fraction expansion of α Acknowledgment The author would like to thank A l a n H a y n e s for his invaluable advice and tutelage, and J e n s M a r k l o f for pointing out the work of Cosentino and Flaminio REFERENCES [] BERESNEVICH, V HAYNES, A VELANI, S: Sums of reciprocals of nα γ) modulo and multiplicative Diophantine approximation, preprint [2] COSENTINO, S FLAMINIO, L: Equidistribution for higher-rank abelian actions on Heisenberg nilmanifolds, JModDyn9 205), [3] HARDY, G H LITTLEWOOD, J E: Some problems of Diophantine approximation a series of cosecants), Bull Calcutta Math Soc ), [4]HARDY,GH LITTLEWOOD,JE PÓLYA, G: Inequalities, Cambridge Mathematical Library 2nd edition), Cambridge University Press, Cambridge, 952 [5] HARMAN, G: Metric Number Theory, In: London Mathematical Society Monographs New Series, Vol 8 The Clarendon Press, Oxford University Press, New York, 998 [6] HAYNES, A: Equivalence classes of codimension one cut-and-project sets, ErgodicTheory Dyn Syst ), no 3, [7] KHINCHIN, A YA: Continued Fractions, The University of Chicago Press, Chicago, III London,

14 CHRISTOPHER J WHITE [8] ROCKETT, A SZÜSZ, P: Continued Fractions, World Scientific Publishing Co, Inc, River Edge, NJ, 992 [9] SINAI, YA G ULCGRAI, C: Estimates from above of certain double trigonometric sums, J Fixed Point Theory Appl ), no, 93 3 Received November, 205 Accepted October 0, 206 Christopher J White School of Mathematics University of Bristol Bristol UK chriswhite@bristolacuk 24