Pustyl'nikov, Lev D.

Schmeling, Jörg

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Institut fiir Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. On some estimations of Weyl sums Lev D. Pusty l 'nikov1 , J org Schmeling2 submitted: 3rd May 1994 1 Electric Power Research Institute VNIIF 2 Institut fiir Angewandte Analysis Kashirskoe shosse 22-3 und Stochastik Moscow 115201 MohrenstraBe 39 Russia D - 10117 Berlin Germany Preprint No. 98 Berlin 1994 Edited by Institut fiir Angewandte Analysis und Stochastik (IAAS) Mohrenstrafie 39 D - 1011 7 Berlin Germany Fax: + 49 30 2004975 e-mail (X.400): c=de;a=d400;p=iaas-berlin;s=preprint e-mail (Internet): preprint@iaas-berlin.d400.de 1. Introduction 2. Definitions and notations 3. Main theorems 4. Concluding remarks 5. Acknowledgements References CONTENTS 1 1 2 6 6 7 1. INTRODUCTION The problem of finding as good as possible estimations of Weyl sums Sp = f e27rif(x), where f ( x )-a polynomial of degree n, (i.e. estimations of the absolute x=l value of such sums) is one of the most important, interesting and hard questions in analysis and number theory. In the present time non-trivial estimations are mainly based on applications of methods of Weyl and Winogradov ( [1], [2], [3], [4]), which give diminishing factors f::l. = p-ik and f::l. = p-~, respectively. The later means that ISPI ::; cP!::l., where c and 'Y are constants depending only on n. The disadvantage of these estimations is that they differ less and less from the trivial one as n tends to infinity, because the exponent in f::l. tends to zero. Therefore, it is of great importance to find such non-trivial estimations of IS p I in which the exponent p in f::l. = p-p (p > 0) won't depend on the degree of the polynomial. In [5] and [6] such estimations are found for all p < ~ and all n greater some number n 0(p) and applied to number-theoretic problems like the estimation of the residual term in the law of the distribution of the fractal parts of the values of a polynomial or their joint distributions. In theorem 5 (§4, chapter 1) in [5] it is proved that assuming the leading coeffi-cient of the polynomial f( x) = a1x + · · · + anxn, an to be irrational and the vector ( { a1}, ... , { an_1}) - consisting of the fractal parts of the other coefficients - can be good approximated by some set of full Lebesgue measure in the (n-1)-dimensional torus Tn-l then ISPI ::; cP1-P for 2 ::; P ::; q, where c and q depend only on the degree of the approximation and not on P. In particular, if a= ( {a1}, ... , {an-1}) belongs to this set of full measure then the estimation ISPI ::; cP1-P is valid for all natural P ~ 2. The present paper directly attaches to [5]. The aim of it is to investigate some of the topological and measure-theoretic properties of the approximation set and related ones. The main results are summarized in theorem 1 and 2. In the first theorem we show that ISPI < pl-p if p < ~ and n > n 0 (p) for arbitrary large intervals of values of P as long as an is irrational and a = ( {al}, ... , { an-l}) belongs to an open and dense subset of Tn-l which has full Lebesque measure. The second theorem shows that it is impossible to find an exponent in the diminishing factor which is valid for all polynomials rather than for a subset of full measure. Namely, there is a - in the topological sense - large set of coefficients such that the corresponding Weyl sums have absolute value abitrary close to P as P tends to infinity. This emphasizes the importance of investigations of the nature of the approximation set and the related approximation process. 2. DEFINITIONS AND NOTATIONS (1) n denotes a natural number greater 3. (2) Tn-l is the ( n - 1 )-dimensional torus considered as Tn-l = {(a1, ... ,an-1)10::; ai < 1 i = 1, ... ,n -1} 1 which is the direct product of ( n - 1) circles of length 1. (3) mes - Lebesque measure on Tn-t . ( 4) For natural k and t the symbol (!) denotes the corresponding binomical coefficient. (5) If an E JR (JR - the real numbers) we define the transformation An-t An-t(an) of Tn-t by: a: = "~t (s: v) (-lta.+v + (n: s) (-1)"-•a,. mod 1 l::;s::;n-1 (6) f(x) denotes the polynomial f(x) = atx + · · · + anxn over the reals with vanishing constant term and irrational leading coefficient an. The vector a= (at, ... , an-t) belongs to Tn-t. (7) For a natural number P we write Sp for the Weyl sum p Sp = L e2?rif(x) x=t (8) p is a positive real number less than ~· (9) no = no(P) = 2 + t~2P. 3. MAIN THEOREMS Theorem 1. Suppose that n > n0 (p) and C is an arbitrary positive number. Then there exist an open subset n = n( C) of the torus Tn-t with properties: (1) The complement Tn-t\n is nowhere dense and has zero Lebesgue measure in Tn-t. (2) For all a= (at, ... , an-t) E f2 there exist a constant Po = P0 (a) such that for Po ::; P ::; Po + C the inequality ISPI::; pt-p holds. Proof. Let Pt be the constant defined by 1 1 Pt= -2 - 1 · no-The definition of n 0 in (9) implies 1 P <Pt< 2 2 (3.1) In the following we use the results of theorem 5, §4, chapter 1 in [1]. For n > no = 1 + 1_;P1 this theorem states the existence of a Lebesgue measur~ble set r n-l = r n-l (p1 ) C Tn-l having the following properties: 00 " " (I) mes (Tn- 1 \ U Ak(r n-1 )) = 0 where Ak is the k-th iteration of the trans-k=O formation A and A0 is the identity. (II) If for some (3 E r n-1 , integers k ~ 0 and q ~ 0 a = ( a1 , ••• , an-1 ) can be written as a=Akf3+z where z = (z1 , ..• , Zn-1 ) is a vector with coordinates fulfilling the inequali-ties lz,I ~ 2:(:~·l) ,s = 1, ... ,n -1 then for all integer P from the interval 2 ::; P ::; q the inequality (3.2) holds. For k = 0, 1, ... we define two sequences Pk and qk of natural numbers and a sequence Uk of subsets of Tn- 1 in the following manner. We set 1 1 Pk= (k + 1 + ((n - l)!)2(n-1) )P1-P and let qk be an arbitrary number fitting the inequality Pk+ G < qk and nk is the union of all open parallelepipedes (3.3) (3.4) where a~ (s = 1, ... ,n -1) are the coordinates of the vector a'= (a~, ... , a~_1 ) = Akf3 and f3 is an arbitrary element of the set r n-1· Now we set 00 n = U nk. k=O We will show that for this set n theorem 1 is true. Obviously n is open and n 00 " contains U Ak(r n-1 ). Therefore, using (I), we have k=O mes (Tn-1 \n) = 0. Moreover this yields that Tn-l \n is nowhere dense in Tn- 1 . Let us assume that for some k a= (ai, ... , an-1 ) E nk. For such an a we set P0 =Pk. The definition of nk then tells us that a can be written as a = a' + z, with a'= Ak(3, (3 E r n-1 3 and z = (z1 , ... , Zn-1 ) respects the inequalities -11-p1 lz.I < 2:(n- l)' s = 1, .. .,n-1. Now (II) implies inequality (3.2) for the polynomial f(x) = a1x + · · · + anxn with a= (a1 , ... , an-1 ) E nk and for 2::; P::; qk. From (3.3) and (3.1) we can derive 1 ISPI ::; (k + 1 + ((n - l)!) 2Cn-l) )Pl-pi :=; pl-p (3.5) for qk ~ P ~ Po = Pk. Now (3.4) implies the theorem. D Next we want to show that the estimates of [5] and [6] of theorem 1 can't be improved. Namely, there is no diminishing factor /J,. = p-p, p > 0, estimating Weyl sums for all polynomials with irrational main coefficient. Definition 3.1. A set is called residual if it contains a countable intersection of open and dense sets. Remark. Residual subsets of Tn are always dense and the intersection of a count-able number of residual sets is residual (see f.i. [7]). Theorem 2. Let n ~ 2. then there exists a residual subset W in Tn such that for a= (a1, ... , an) E W, f(x) = alx + · · · + anxn and arbitrary p > 0 the inequality ISPI ::; pl-p is violated infinitely often. The proof is based on some well-known facts about Weyl-sums (Lemma 1-3) which one can find f.i. in [4]. Lemma 1. Assume that J( x) = f 1 ( x )+ · · ·+ f 11 ( x) and the fractal parts of fi( x ), i = 1, ... , s are periodic with mutually relatively prime periods r1 , •.. , r11 • Then Lemma 2. Assume that 2a2 and q are relatively prime then: 4 Lemma 3. Assume that a and p are mutually prime and n 2:: 3 then: Let us fix n 2:: 2 and consider the sets Wm= {a E Tnl3P 2:: m such that ISPI > P1-~} where Sp = Sp(a) = E e21ri(a1:z:+··+an:r:n), m E N. Because finite Weyl sums :r:=l depend continuously on a E Tn, all the sets Wm, m E N are open. Lemma 4. Wm,m EN, is dense in Tn. Proof. We fix e > 0 and a = (a1, ... , an)· Subject to lemma 2 we select integers ai, a2 and q2 with la1l < q2, la2l < q2, q2 - an odd prime number, I ai - a 1 I < e and q2 la2 - a2l < e. q2 For 2 < s :::; n we choose an integer as and a - prime to as - prime number qs, qs > qs-1 with Hence the polynomial f ( x) ~x + !!2.x2 + ~x3 + · · · + ~xn is e-close in the q2 q2 qg q~ topology induced from Tn to f(x) = a 1x + · · · + anxn and according to lemma 1-3 It is easy to see that for k E N 1sk·q2q:· ... ·q~ I = k 1sq2q:-. .. ·q~ I = k . s. Setting q2 • q~ · ... · q;: = q and Pk= kq we can derive ISP1cl = ISkql = k. s > (kq)1-~ = p:-~ if k > 2;,:1 • Moreover, if P = kq + r, 0:::; r < q and k > 2;,:1 + T then ISPI = ISkq+rl > k · S - r > (kq + r)1-~ = p1-~. This means a = ( ~, !!a, ~, ..• , ~) belongs to Wm for all m E N, and consequently q2 q2 q3 qn all Wm are dense, because of the arbitrary choice of e. D 5 Proof of the theorem: We define 00 m=O If a point a E Tn belongs to W then it has the property: there are two sequences {Pi} and {mz} of natural numbers, such that ISPz I > Pz(l- ~z) But this gives the statement of theorem 2. D Corollary 3.1. The set of vectors a= (a1, ... , an) E Tn with all ai (i = 1, ... ,n) irrational and violating ISpl < pl-p for infinitely many P and arbitrary p > 0 is residual. Proof. The set W = W n {(S1\Q) x · · · x (S1\Q)} n-times fits the theorem. the set {(S1 \Q) x · · · x (S1\Q)} is residual and hence by the remark following definition 3.1. W is residual itself. D 4. CONCLUDING REMARKS In [4] it is shown, that we can get uniform estimates (not depending on the degree of the polynomial) of the Weyl sum ISPI for all large enough P for a set of coefficients 3 in Tn-l having full Lebesque measure. Moreover, these estimates are depending on the approximability of the vector ( a1 , •.• , an-l) by a certain approximation process (see point II). On the other hand the corollary to theorem 2 shows that the complement of 3 in Tn-l is residual. These facts suggest that a more precise description of that approximation process together with a measure-theoretical and topological analysis of sets related to that process could help to get a much better understanding of Weyl sums and related problems. 5. ACKNOWLEDGEMENTS This work was prepared during a visit of L.D. Pustyl'nikov to the IAAS. L.D. Pustyl 'nikov thanks the IAAS for supporting his stay in Berlin. 6 REFERENCES 1. H. Weyl: Uber die Gleichverteilung der Zahlen mod 1., Math. Ann. 77 (1916), 313-.352. 2. K. Chandrasekharan: Arithmetical Functions, Springer, 1970. 3. l.M. Vinogradov: The Method of Ezponential Sums in the Theory of Numbers, Nauka, Moscow, 1971. 4. N.M. Korobov: Ezponential Sums and their applications, Nauka, Moscow, 1989. 5. L.D. Pustyl'nikov: The distribution of the fractional parts of a polynomial's values, Weyl sums and ergodic theory, Uspekhi Mat. Nauk 4 (1993), 131-166. 6. L.D. Pustyl'nikov: New estimates of Weyl sums and the remainder term in the law of dis-tribution of the fractional part of a polynomial, Ergodic Theory and Dynamical Systems, 11 (1991), 515-534. 7. J.C. Oxtoby: Measure and Category, Springer 1980 7 Recent publications of the Institut fiir Angewandte Analysis und Stochastik Preprints 1993 69. John W. Barrett, Peter Knabner: Finite element approximation of transport of reactive solutes in porous media. Part 2: Error estimates for equilibrium adsorption processes. 70. Herbert Gajewski, Willi Jager, Alexander Koshelev: About loss of regularity and "blow up" of solutions for quasilinear parabolic systems. 71. Friedrich Grund: Numerical solution of hierarchically structured systems of algebraic-differential equations. 72. Henri Schurz: Mean square stability for discrete linear stochastic systems. 73. Roger Tribe: A travelling wave solution to the Kolmogorov equation with noise. 74. 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On some estimations of Weyl sums

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On some estimations of Weyl sums

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