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ON THE BOMBIERI-KOROBOV ESTIMATE FOR WEYL SUMS1 SCOTT T. PARSELL2 1. Introduction In studying Waringβs problem and other additive questions involving πth powers, a fundamental role is played by estimates for the exponential sum β π (πΆ) = ππ (πΆ; π ) = π(πΌπ π₯π + β β β + πΌ1 π₯), 1β€π₯β€π 2πππ¦ where we have written π(π¦) = π . When πΌπ lies in a suitable set of minor arcs, methods based on Weyl diο¬erencing and Vinogradovβs mean value theorem yield non-trivial upper bounds for β£π (πΆ)β£. In Vinogradovβs method, which is more eο¬ective for larger π, one obtains minor-arc estimates for β£π (πΆ)β£ from bounds for the mean values β« π½π ,π (π ) = β£π (πΆ)β£2π ππΆ. [0,1]π When π is an integer, one sees by orthogonality that π½π ,π (π ) is the number of solutions of the system of equations π₯π1 + β β β + π₯ππ = π¦1π + β β β + π¦π π (1 β€ π β€ π) (1.1) with π₯π , π¦π β [1, π ] β© β€. By using a π-adic iteration method, one may obtain estimates of the shape 1 π½π ,π (π ) βͺ π 2π β 2 π(π+1)+π(π ,π)+π , (1.2) where the sharpest values for π(π , π) are due to the repeated eο¬cient diο¬erencing technology of Wooley [9]. We observe that the mean value π½π ,π (π ) is well-deο¬ned for non-integral π as well. Although it then lacks an interpretation as the number of solutions of (1.1), estimates of the shape (1.2) follow by interpolation via HoΜlderβs inequality. When πΌπ has a rational approximation with relatively large denominator, arguments of Bombieri [2] and Korobov [5] produce estimates of the shape β£π (πΆ)β£ βͺ π 1βπ(π)+π , where π(π) is determined by the available mean value estimates. This technique, when combined with the results of Wooley [11] on smaller moduli, yields the sharpest currently available Weyl exponents for π β₯ 14. The main purpose of this note is to provide an alternative method for obtaining the results for larger moduli. In contrast to the arguments of Bombieri and Korobov, our derivation makes use of a standard Weyl shift and the large sieve in a manner familiar to additive number theorists. In the process, we make an additional observation that yields modest improvements in the existing Weyl exponents. In the standard 2000 Mathematics Subject Classiο¬cation. Primary 11L15; Secondary 11L07, 11P55. Key words and phrases. Weyl sums, exponential sums, Hardy-Littlewood method. 1 There is a confusing misprint in the published version of this paper. In the statement of Theorem 1.1 and also later on page 368, the quantity π½π/2,π‘ (2π ) should be replaced by πΌπ,π‘ (2π ). The surrounding arguments and results are unaο¬ected. 2 Supported by a National Security Agency Young Investigator Grant (MSPF-07Y-221). 1 2 SCOTT T. PARSELL arguments (see for example [1], chapter 4, and [7], chapter 5) the Weyl shift initially makes use of a rather arbitrary set of integers within the set [1, π ], which is later specialized in order to achieve the spacing condition required for an application of the large sieve. In our argument, we retain the arbitrary nature of the set and simply split the relevant sum into a bounded number of sub-sums, each having the desired property. Moreover, because of an interchange in the order of summation, we obtain an eο¬ect similar to that of Wooley [11], Lemma 4, in that the main mean values relevant to our argument involve variables restricted to this set. Thus we can potentially gain an advantage by using a convenient set of integers. The same principle may be applied to the arguments of [1] and [7] by interchanging the roles of the variables. We ο¬rst state a result essentially equivalent to Theorem 8 of Bombieri [2]. Theorem 1.1. Let π β [1, π ] β© β€ with β£πβ£ = π, and write π½π ,π (π) for the number of solutions of (1.1) with π₯π , π¦π β π. If β£ππΌπ β πβ£ β€ π β1 with (π, π) = 1 and π, π , and π‘ are positive integers with π‘ β€ π β 1, then one has 1 β£π (πΆ)β£2ππ βͺ πβ2π (log 2π )2ππ π»(π, π )π 2ππ βπ+ 2 π(πβ1) πΌπ,π‘ (2π )π½π ,πβ1 (π), where π»(π, π ) = πβ1 β (1 + ππ βπ )(1 + π πβπ π β1 ) π=πβπ‘ and πΌπ,π‘ (π ) denotes the maximum over n of the number of solutions of the system π₯π1 + β β β + π₯ππ = ππ (1 β€ π β€ π‘) with x β [1, π ]π . We now describe an interesting special case of the theorem. Let π(π, π ) = {π β [1, π ] β© β€ : πβ£π, π prime β π β€ π } denote the set of π -smooth numbers up to π , and let ππ ,π (π, π ) denote the number of solutions of the system (1.1) with π₯π , π¦π β π(π, π ). When π is a suο¬ciently small power of π , the technology of Wooley [13] yields estimates of the shape 1 ππ ,π (π, π ) βͺ π 2π β 2 π(π+1)+Ξ(π ,π)+π , (1.3) where Ξ(π , π) behaves essentially like π(π , π) for large π but may in fact be somewhat smaller for speciο¬c moderately-sized values of π. We sometimes refer to exponents π(π , π) and Ξ(π , π) satisfying (1.2) and (1.3) as admissible. The advantages of using smooth numbers would be much more signiο¬cant if we were dealing with an incomplete system, but we can nevertheless expect some modest improvements. Corollary 1.2. Suppose that β£ππΌπ β πβ£ β€ π β1 , where π π β€ π β€ π πβπ and (π, π) = 1. Whenever 1 β€ π β€ π/2 and Ξ(π , π β 1) is an admissible exponent for (1.3), one has β£π (πΆ)β£ βͺ π 1βπ(π)+π , where π β Ξ(π , π β 1) . π(π) = 2ππ One may now optimize over π in conjunction with Lemma 4 of Wooley [11] to obtain new Weyl exponents. We give a description of these computations in §4. The author thanks Craig Spencer, Bob Vaughan, and Trevor Wooley for helpful discussions. ON THE BOMBIERI-KOROBOV ESTIMATE FOR WEYL SUMS1 3 2. Preliminaries We begin by recalling a standard Weyl shift (see equation (5.23) of Vaughan [7]). Lemma 2.1. Let π β [1, π ] β© β€ with β£πβ£ = π. Then one has β β£π(πΆ; π½, π¦)β£, β£π (πΆ)β£ βͺ πβ1 log 2π sup π½β[0,1] π¦βπ where ) 2π (β π β π π(πΆ; π½, π¦) = π πΌπ (π₯ β π¦) + π½π₯ . π₯=1 π=1 We also need the following multi-dimensional version of the large sieve (see Vaughan [7], Lemma 5.3). Lemma 2.2. Suppose that Ξ β βπ and that the sets π (πΈ) = {π· : β£β£π½π β πΎπ β£β£ < πΏπ , 0 β€ π½π < 1} with πΈ β Ξ are pairwise disjoint. Let β³ denote the set of integer π-tuples m with 1 β€ ππ β€ ππ , and write β π(π·) = π(m)π(π· β m). mββ³ Then one has β 2 β£π(πΈ)β£ βͺ πΈβΞ β β£π(m)β£ 2 π β (ππ + πΏπβ1 ). π=1 mββ³ Finally, we recall a result on the number of solutions of a diophantine inequality (see Bombieri [2], Lemma 3). Lemma 2.3. If πΌ is a real number with β£ππΌ β πβ£ β€ π β1 for some integers π and π with (π, π) = 1 and π is a positive integer, then the number of solutions of the inequality β£β£ππΌπ₯ + π½β£β£ β€ 1/π with β£π₯β£ β€ π and π₯ β β€ does not exceed (1 + 4π/π )(1 + 4ππ/π). 3. The alternative proof In this section we obtain the Weyl-type estimates advertised in Theorem 1.1. Suppose that β£ππΌπ β πβ£ β€ π β1 , where (π, π) = 1, let π be a set of integers in [1, π ] with β£πβ£ = π, and let π, π , and π‘ be positive integers with π‘ β€ π β 1. Then by Lemma 2.1 and HoΜlderβs inequality we have ¯ )¯¯π 2π (β π β ¯¯β ¯ πΌπ (π₯ β π¦)π + π½π₯ ¯ π β£π (πΆ)β£π βͺ πβ1 πΏπ ¯ ¯ ¯ π₯=1 π=1 π¦βπ for some π½ β [0, 1], where we have written πΏ = log 2π . Hence there exist complex numbers ππ¦ with β£ππ¦ β£ = 1 such that ) π β β (β π β1 π β£π (πΆ)β£ βͺ π πΏ ππ¦ π πΌπ ππ (x, π¦) + π½(π₯1 + β β β + π₯π ) , π¦βπ xβ[1,2π ]π π=1 4 SCOTT T. PARSELL where we have written ππ (x, π¦) = (π₯1 β π¦)π + β β β + (π₯π β π¦)π . Now by interchanging the order of summation and using HoΜlderβs inequality again we obtain ¯ (β )¯¯2π π β ¯¯β ¯ β£π (πΆ)β£2ππ βͺ πβ2π πΏ2ππ π π(2π β1) ππ¦ π πΌπ ππ (x, π¦) ¯ . ¯ ¯ ¯ π xβ[1,2π ] We next observe that π β π π π πΌπ (π₯ β π¦) = (β1) πΌπ π¦ + π=1 π¦βπ πβ1 β π=1 π π΄π (π₯)π¦ + π=1 where π π΄π (π₯) = (β1) π=π πΌ π π₯π , π=1 π ( ) β π π π β πΌπ π₯πβπ . We therefore deduce that β£π (πΆ)β£2ππ ¯ ( )¯¯2π πβ1 β ¯¯β β ¯ βͺΞ₯ π΅π (x)π¦ π ¯ , ππ¦ π (β1)π ππΌπ π¦ π + ¯ ¯ ¯ π xβ[1,2π ] π=1 π¦βπ where Ξ₯ = πβ2π πΏ2ππ π π(2π β1) and π ( ) β π πΌπ (π₯1πβπ + β β β + π₯ππβπ ). π΅π (x) = (β1) π π=π π Let π(n) denote the number of solutions of the system π₯π1 + β β β + π₯ππ = ππ (1 β€ π β€ π‘) π with x β [1, 2π ] , and write π πΎπ (n) = (β1) π ( ) β π π=π π πΌπ ππβπ . Further let π(m) denote the number of solutions of the system π¦1π + β β β + π¦π π = ππ (1 β€ π β€ π β 1) with y β ππ , and let πΛ(m) denote the corresponding weighted count, where each solution is counted with weight ) ( π€y = ππ¦1 β β β ππ¦π π (β1)π ππΌπ (π¦1π + β β β + π¦π π ) . We then have β£π (πΆ)β£2ππ ¯ )¯¯2 (β πβ1 πβπ‘β1 β β ¯¯β ¯ π΅π (x)ππ ¯ , π€y π πΎπ (n)ππ + βͺΞ₯ ¯ ¯ ¯ n,x m,y π=πβπ‘ π=1 where the summations are over x counted by π(n) and y counted by π(m). Write ( ) ( ) ( ) πβ1 πβπ‘β1 β β πβπ‘ π πβπ‘ π= and π2 = π= β , π1 = 2 2 2 π=1 π=πβπ‘ ON THE BOMBIERI-KOROBOV ESTIMATE FOR WEYL SUMS1 5 and view m = (m1 , m2 ) as an element of β€πβπ‘β1 × β€π‘ . Using Cauchyβs inequality to isolate the contribution from the indices π with π β π‘ β€ π β€ π β 1, we obtain ¯ )¯¯2 (β πβ1 β β ¯¯β ¯ β£π (πΆ)β£2ππ βͺ Ξ₯ π(n)π π1 πΎπ (n)ππ ¯ . (3.1) πΛ(m)π ¯ ¯ ¯ n m m 1 π=πβπ‘ 2 Now for each π with π β π‘ β€ π β€ π β 1 we have πΎπ (n) = ππ πΌπ ππβπ + ππ (n) for some integer ππ , where ππ (n) depends only on ππβπβ1 , . . . , π1 and πΆ. Hence we may successively apply Lemma 2.3, starting with π = π β 1, to deduce that the system of diophantine inequalities β£β£πΎπ (n) β πΎπ (nβ² )β£β£ < π βπ (π β π‘ β€ π β€ π β 1) has at most πβ1 β π» = π»(π, π ) βͺ (1 + ππ βπ )(1 + π πβπ π β1 ) π=πβπ‘ solutions n for every ο¬xed nβ² . We may therefore split the sum over n into π» sub-sums over sets π©1 , . . . , π©π» with the property that for each π the sets {π· β [0, 1)π‘ : β£β£πΎπ (n) β π½π β£β£ < 12 π βπ , π β π‘ β€ π β€ π β 1} corresponding to distinct n β π©π are pairwise disjoint. It therefore follows from Lemma 2.2 that for each m1 and each π one has ¯ (β )¯¯2 πβ1 πβ1 β ¯¯β β β ¯ 2 πΎπ (n)ππ ¯ βͺ πΛ(m)π π π. (3.2) β£Λ π(m)β£ ¯ ¯ ¯ nβπ©π m2 m2 π=πβπ‘ π=πβπ‘ On inserting (3.2) into (3.1), we obtain β£π (πΆ)β£2ππ βͺ πβ2π πΏ2ππ π»(π, π )π π(2π β1)+π1 +π2 πΌπ,π‘ (2π ) β β£Λ π(m)β£2 . m Moreover, one has β£Λ π(m)β£ β€ π(m) and β π(m)2 = π½π ,πβ1 (π), m so on noting that π1 + π2 = 21 π(π β 1) we ο¬nd that 1 β£π (πΆ)β£2ππ βͺ πβ2π πΏ2ππ π»(π, π )π 2π πβπ+ 2 π(πβ1) πΌπ,π‘ (2π )π½π ,πβ1 (π), and this completes the proof of Theorem 1.1. In order to deduce Corollary 1.2, we take π‘ = π and π = π(π, π ) with π a small power of π , so that π β π . By applying the argument of the proof of Vaughan [7], Lemma 5.1, one sees that πΌπ,π (2π ) β€ π!. Moreover, the hypothesis that π π β€ π β€ π πβπ with π β€ π/2 shows that π»(π, π ) βͺ 1. The corollary now follows immediately on substituting the estimate 1 π½π ,πβ1 (π) = ππ ,πβ1 (π, π ) βͺ π 2π β 2 π(πβ1)+Ξ(π ,πβ1)+π . One could also reverse the roles of the variables in the Weyl shift (see Wooley [11], Lemma 2) and then attempt to take π > π‘ and inject smooth number technology into the analogue of π(n), but this seems to produce inferior results. 6 SCOTT T. PARSELL 4. An application As mentioned in the introduction, the ο¬exibility arising from the set π in Theorem 1.1 permits modest improvements on the Weyl exponents calculated by Ford [3]. Our aim in this section is to brieο¬y describe the ingredients involved in using Corollary 1.2 to carry out such computations. As a starting point for our mean value estimates, we recall that π½π ,π (π ) and ππ ,π (π, π ) exhibit diagonal behavior for π β€ π + 1 (see Hua [4], Lemma 5.4). Hence the exponents π(π , π) = Ξ(π , π) = 21 π(π + 1) β π are admissible when π β€ π + 1. We obtain further admissible exponents for (1.3) using the iterative method of Wooley [13]. Lemma 4.1. Suppose that Ξπ = Ξ(π , π) is an admissible exponent for (1.3). Given π with 1 β€ π β€ π, deο¬ne ππ = 1/π and for π½ = π, . . . , 2, set ) ( 1 π½(π½ β 1) β 2Ξπ 1 β ππ½ + + ππ½β1 = 2π 2 4π(π β π½ + 1) and ππ½β1 = min{1/π, πβπ½β1 }. Then the exponent Ξπ +π = Ξπ (1 β π1 ) + π(ππ1 β 1) is also admissible. Proof. This is a special case of [13], Lemma 6.1. Here one has π‘ = π, ππ = π β π + 1, in the notation of that lemma. πΛπ½ = π β π½ + 1, 1 and Ξ©π½ = π½(π½ + 1) 2 β‘ Note that the parameter π1 depends on the number of diο¬erences π (in addition to π ), so for each π we minimize over 1 β€ π β€ π to obtain the optimal value. As mentioned in [10], the methods of [13] also permit one to establish quasi-diagonal behavior for the system (1.1) restricted to smooth numbers. The fundamental relationship is embodied in the following lemma. Lemma 4.2. Let π = βπ/2β, and suppose that π and π‘ are natural numbers satisfying 3 β€ π β€ π, 1 β€ π‘ < 2π, and π + π‘ β₯ π. If π£ = π (1 β π‘/2π)β1 and 0 < π β€ 1/π then one has ( ) ππ +π‘,π (π, π ) βͺ π (2π +π(π,π‘,π))π π π‘ ππ ,π (π, π ) + π (π‘/2)(2βππ) (ππ£,π (π, π ))π /π£ , where π = π 1βπ and 1 π(π, π‘, π) = (π + π‘ β π β 1)(π + π‘ β π). 2 Proof. This follows by applying the arguments of [13], Lemmas 4.1 and 4.2, within the argument of [10], Lemma 4.2, after adjusting the application of HoΜlderβs inequality in the latter proof to accommodate a fractional moment. β‘ If one has estimates of the shape ππ ,π (π, π ) βͺ πππ +π , then the two terms in Lemma 4.2 are equal when 2(π ππ£ β π£ππ ) . π= 2π ππ£ + ππ‘π£ β 2π£ππ We take this value for π whenever it does not exceed 1/π and take π = 1/π otherwise. As in the computations of Ford [3], we apply Lemmas 4.1 and 4.2 iteratively to obtain the best results. We may then apply the argument of Wooley [11], Theorem 2, using Corollary 1.2 in place of [11], Lemma 3, and optimize over 1 β€ π β€ π/2 to obtain new ON THE BOMBIERI-KOROBOV ESTIMATE FOR WEYL SUMS1 7 Weyl exponents. For larger π, our improvements are not signiο¬cant, as the advantage of restricting to smooth numbers is not overwhelming when already considering the complete Vinogradov-type system After obtaining some preliminary admissible exponents, one can apply the method of Baker [1], chapter 4, to generate Weyl-type estimates on a set of minor arcs in [0, 1]π . By employing a suitable Hardy-Littlewood dissection, these estimates can be used in conjunction with the preliminary Vinogradov exponents and standard major arc information to show that π(π , π) = 0 and Ξ(π , π) = 0 are admissible whenever π is suο¬ciently large in terms of π. Improved admissible exponents can then be calculated by further iterating the above lemmas. Lemma 4.3. Suppose that Ξ(π , π β 1) is admissible for (1.3), and let 1 β 2Ξ(π , π β 1) . (πβ1)β£π 4π π (π) = max If β£π (πΆ)β£ β₯ π 1βπ½(π) , where π½(π) < max{21βπ , π (π)}, then there exist integers π, π1 , . . . , ππ with (π, π1 , . . . , ππ ) = 1 such that π β€ π 1/π and β£ππΌπ β ππ β£ β€ π 1/πβπ (1 β€ π β€ π). (4.1) Proof. This follows from Theorem 5.1 and the proof of Theorems 4.3 and 4.4 in Baker [1] upon reversing the roles of the variables in the Weyl shift. β‘ Let π denote the subset of [0, 1]π for which there exist π and a with (π, π1 , . . . , ππ ) = 1 satisfying (4.1). Then a standard major arc analysis along the lines of Vaughan [7], chapter 7 (see also [8], Lemma 9.2 and [6], Lemmas 3.2 and 3.3) establishes the following. Lemma 4.4. Whenever π β₯ 12 π(π + 1), one has β« 1 β£π (πΆ)β£2π ππΆ βͺ π 2π β 2 π(π+1)+π π and whenever π > 1 π(π 2 β« + 1), one has 1 β£π (πΆ)β£2π ππΆ = π(π , π)π 2π β 2 π(π+1) (1 + π(π βπΏ(π) )). (4.2) π for some real numbers π(π , π) and πΏ(π) > 0. Here the number π(π , π) is the product of the expected singular integral and singular series, and its positivity actually follows directly from (4.2) in view of the elementary lower 1 bound π½π ,π (π ) β« π 2π β 2 π(π+1) , provided that π is large enough to show that the contribution from the minor arcs has smaller order of magnitude. After infusing smooth mean values into Lemma 4 of Wooley [11], we are ο¬nally able to describe some small improvements on existing Weyl exponents for π β₯ 11. Theorem 4.5. Suppose that there are coprime integers π and π with π β€ π β€ π πβ1 such that β£ππΌπ β πβ£ β€ π β1 . Then one has β£π (πΆ)β£ βͺ π 1βπ(π)+π , where π(π) β₯ 1/π1 (π) and π1 (π) is given in Table 4.1. When π = 1, Corollary 1.2 suο¬ces to cover a complete set of minor arcs, and the resulting estimate turns out to be optimal when 11 β€ π β€ 13. As in the work of Ford [3], π = 2 is optimal for 14 β€ π β€ 20. Further computations reveal that π = 2 continues to be optimal 8 SCOTT T. PARSELL up to π = 24, while π = 3 becomes optimal starting at π = 25. Combining these values Λ with Lemma 5.4 of Ford [3], we obtain the upper bounds πΊ(π) β€ π»1 (π) for the values of π»1 recorded in Table 4.1. Table 4.1. Weyl exponents and the asymptotic formulas π 11 12 13 14 15 16 17 18 19 20 π1 (π) π»1 (π) 743.409 706 999.270 873 1223.475 1049 1420.574 1231 1632.247 1431 1856.535 1645 2114.819 1879 2436.255 2134 2779.680 2410 3150.605 2701 π2 (π) π»2 (π) 1489.646 504 2027.215 661 2642.870 803 3232.507 955 3834.340 1120 4501.372 1299 5209.886 1492 5988.000 1699 6815.154 1922 7705.730 2158 Admissible values for the π½(π) arising in Lemma 4.3 satisfy π½(π) β₯ 1/π2 (π) for the values of π2 recorded in Table 4.1. These may then be used in conjunction with the Vinogradov exponents and Lemma 4.4 to produce asymptotic formulas for π½π ,π (π ) as in the argument of Wooley [12], Theorem 3. Let πΉΛ(π) denote the least integer π for which π½π ,π (π ) has an asymptotic formula of the shape given on the right-hand side of (4.2). Then one has the upper bounds πΉΛ(π) β€ π»2 (π) for the values of π»2 recorded in Table 4.1. References [1] R. C. Baker, Diophantine inequalities, Clarendon Press, Oxford, 1986. [2] E. Bombieri, On Vinogradovβs mean value theorem and Weyl sums, Proceedings of the conference on automorphic forms and analytic number theory, Univ. Montreal, 1990, pp. 7β24. [3] K. B. Ford, New estimates for mean values of Weyl sums, Internat. Math. Res. Notices (1995), 155β 171. [4] L.-K. Hua, Additive theory of prime numbers, A.M.S., Providence, Rhode Island, 1965. [5] N. M. Korobov, Weylβs estimates and the distribution of primes, Dokl. Akad. Nauk SSSR 123 (1958), 28β31. [6] S. T. Parsell, On simultaneous diagonal inequalities, III, Quart. J. Math. 53 (2002), 347β363. [7] R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge University Press, Cambridge, 1997. [8] T. D. Wooley, On simultaneous additive equations II, J. Reine Angew. Math. 419 (1991), 141β198. [9] , On Vinogradovβs mean value theorem, Mathematika 39 (1992), 379β399. [10] , Quasi-diagonal behaviour in certain mean value theorems of additive number theory, J. Amer. Math. Soc. 7 (1994), 221β245. , New estimates for Weyl sums, Quart. J. Math. Oxford (2) 46 (1995), 119β127. [11] , Some remarks on Vinogradovβs mean value theorem and Tarryβs problem, Monatsh. Math. [12] 122 (1996), 265β273. , On exponential sums over smooth numbers, J. Reine Angew. Math. 488 (1997), 79β140. [13] Department of Mathematics, West Chester University, 25 University Ave., West Chester, PA 19383, U.S.A., [email protected]