Multiple exponential sums over smooth numbers. (English) Zbl 1021.11024
In this substantial paper, estimates for multiple exponential sums are developed and applied to a variety of problems.
One of the author’s goals is to establish the existence of non-trivial rational lines on hypersurfaces defined by the vanishing of a diagonal form: \(\sum_{1 \leq i \leq s}c_ix_i^k\), subject to certain conditions, one of which should be that the number \(s\) of variables in the equation need not exceed a realistically small value. In this respect his theorem improves substantially on an approach developed by R. Brauer and later B. J. Birch, which does not yield a reasonably small bound, and guarantees only the existence of linear spaces on the hypersurfaces which, although of reasonably small degree, cannot be guaranteed to be lines. Relatively recently, some calculations of the number of variables needed in an improved approach of this type in the cubic case have been performed by T. D. Wooley [Bull. Lond. Math. Soc. 29, 556-562 (1997; Zbl 0910.11013)].
The author uses the Hardy-Littlewood method, which also leads to other applications of a type more commonly seen in this context. Let \(S_s(P)\) denote the number of solutions of the system \[ \sum_{1 \leq m \leq s}x_m^iy_m^j - \overline x_m^i\overline y_m^j=0\;\text{ for} 0 \leq i \leq k\;\text{ and} i+j=k \] in numbers not exceeding \(P\). Then, provided \(s \geq s_1\) for a certain \(s_1\) not very different from \(k^2\), the author shows that \(S_s(P) \ll P^{4s-k(k+1)+\Delta_s+\varepsilon}\) with an “admissible exponent” not exceeding \(k(k+1)\) and decreasing as \(s\) increases. The author can restrict his variables to be \(R\)-smooth (having no prime factors exceeding \(R\)) with \(R \leq P^\eta\), where \(\eta\) depends on \(\varepsilon\). For large \(k\) the result is sharpened to a form with improved admissible exponents by a process of repeated efficient differencing. These results also lead in a standard way to estimates of Weyl sums over minor arcs.
In the related problem in which the restriction \(i+j=k\) is absent G. I. Arkhipov, A. A. Karatsuba and V. N. Chubarikov [Tr. Mat. Inst. Steklova 151 (1980; Zbl 0441.10037); translation in Proc. Steklov Inst. Math. 151 (1982)] had an admissible exponent more like \(k^3\) and needed \(s_1\) of a size like \(6k^3\log k\).
The author also considers a multidimensional analogue of Waring’s problem, introduced by G. I. Arkhipov and A. A. Karatsuba [Sov. Math., Dokl. 36, No. 1, 75-77 (1987); translation from Dokl. Akad. Nauk SSSR 295, 521-523 (1987; Zbl 0655.10043)]. Here he seeks the number of solutions of a system \[ \sum_{1 \leq m \leq s}x_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq k \] with the integers \(x_m,y_m\) in \([1,P]\). A condition on the numbers \(n_j\) has to be imposed, sufficient to ensure that real solutions exist. In addition, a \(p\)-adic condition of the usual type is required. Then \(\gg P^{2s-k(k+1)}\) solutions are found provided \(s \geq s_1\), for a certain \(s_1\) somewhat larger than \({14\over 3}k^2\log k\).
In a related way the author produces a similar result for the system of equations \[ \sum_{1 \leq m \leq s}c_mx_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq k, \] or equivalently, via the binomial theorem, of the equation \(\sum_{1 \leq m \leq s} c_m(x_mt+y_m)^k=0\). The connection with the problem of finding lines on a surface is not immediate, because of the possibility that many of the solutions that have been counted might correspond to the same line. This obstacle is overcome by producing an estimate when the variables are restricted to dyadic intervals, and showing that the number of solutions corresponding to a given line is bounded. In this way the author finds, subject to the usual solubility conditions, \(\gg P^{2s-k(k+1)}\) distinct rational lines on the surface \(\sum_{1 \leq m \leq s}c_mz_m^k =0\) with \(\max\{|x_m|,|y_m|\}\leq P\).
The author announces that in a separate publication [Trans. Am. Math. Soc. 352, 5045-5062 (2000; Zbl 1108.11302)] it is shown that in the cubic case such a result holds unconditionally whenever \(s \geq 57\).
One of the author’s goals is to establish the existence of non-trivial rational lines on hypersurfaces defined by the vanishing of a diagonal form: \(\sum_{1 \leq i \leq s}c_ix_i^k\), subject to certain conditions, one of which should be that the number \(s\) of variables in the equation need not exceed a realistically small value. In this respect his theorem improves substantially on an approach developed by R. Brauer and later B. J. Birch, which does not yield a reasonably small bound, and guarantees only the existence of linear spaces on the hypersurfaces which, although of reasonably small degree, cannot be guaranteed to be lines. Relatively recently, some calculations of the number of variables needed in an improved approach of this type in the cubic case have been performed by T. D. Wooley [Bull. Lond. Math. Soc. 29, 556-562 (1997; Zbl 0910.11013)].
The author uses the Hardy-Littlewood method, which also leads to other applications of a type more commonly seen in this context. Let \(S_s(P)\) denote the number of solutions of the system \[ \sum_{1 \leq m \leq s}x_m^iy_m^j - \overline x_m^i\overline y_m^j=0\;\text{ for} 0 \leq i \leq k\;\text{ and} i+j=k \] in numbers not exceeding \(P\). Then, provided \(s \geq s_1\) for a certain \(s_1\) not very different from \(k^2\), the author shows that \(S_s(P) \ll P^{4s-k(k+1)+\Delta_s+\varepsilon}\) with an “admissible exponent” not exceeding \(k(k+1)\) and decreasing as \(s\) increases. The author can restrict his variables to be \(R\)-smooth (having no prime factors exceeding \(R\)) with \(R \leq P^\eta\), where \(\eta\) depends on \(\varepsilon\). For large \(k\) the result is sharpened to a form with improved admissible exponents by a process of repeated efficient differencing. These results also lead in a standard way to estimates of Weyl sums over minor arcs.
In the related problem in which the restriction \(i+j=k\) is absent G. I. Arkhipov, A. A. Karatsuba and V. N. Chubarikov [Tr. Mat. Inst. Steklova 151 (1980; Zbl 0441.10037); translation in Proc. Steklov Inst. Math. 151 (1982)] had an admissible exponent more like \(k^3\) and needed \(s_1\) of a size like \(6k^3\log k\).
The author also considers a multidimensional analogue of Waring’s problem, introduced by G. I. Arkhipov and A. A. Karatsuba [Sov. Math., Dokl. 36, No. 1, 75-77 (1987); translation from Dokl. Akad. Nauk SSSR 295, 521-523 (1987; Zbl 0655.10043)]. Here he seeks the number of solutions of a system \[ \sum_{1 \leq m \leq s}x_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq k \] with the integers \(x_m,y_m\) in \([1,P]\). A condition on the numbers \(n_j\) has to be imposed, sufficient to ensure that real solutions exist. In addition, a \(p\)-adic condition of the usual type is required. Then \(\gg P^{2s-k(k+1)}\) solutions are found provided \(s \geq s_1\), for a certain \(s_1\) somewhat larger than \({14\over 3}k^2\log k\).
In a related way the author produces a similar result for the system of equations \[ \sum_{1 \leq m \leq s}c_mx_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq k, \] or equivalently, via the binomial theorem, of the equation \(\sum_{1 \leq m \leq s} c_m(x_mt+y_m)^k=0\). The connection with the problem of finding lines on a surface is not immediate, because of the possibility that many of the solutions that have been counted might correspond to the same line. This obstacle is overcome by producing an estimate when the variables are restricted to dyadic intervals, and showing that the number of solutions corresponding to a given line is bounded. In this way the author finds, subject to the usual solubility conditions, \(\gg P^{2s-k(k+1)}\) distinct rational lines on the surface \(\sum_{1 \leq m \leq s}c_mz_m^k =0\) with \(\max\{|x_m|,|y_m|\}\leq P\).
The author announces that in a separate publication [Trans. Am. Math. Soc. 352, 5045-5062 (2000; Zbl 1108.11302)] it is shown that in the cubic case such a result holds unconditionally whenever \(s \geq 57\).
Reviewer: George Greaves (Cardiff)
MSC:
| 11L07 | Estimates on exponential sums |
| 11P05 | Waring’s problem and variants |
| 11D45 | Counting solutions of Diophantine equations |
| 11P55 | Applications of the Hardy-Littlewood method |
References:
| [1] | Arkhipov G. I., Soviet Math. Dokl. 36 pp 75– (1988) |
| [2] | Arkhipov G. I., Trudy Mat. Inst. Steklov 151 pp 1– (1980) |
| [3] | B., Mathematika 4 pp 102– (1957) |
| [4] | Bull. Amer. Math. Soc. 51 pp 749– (1945) |
| [5] | H. Davenport, Analytic methods for diophantine equations and diophantine inequalities, Ann Arbor Publishers, 1962. |
| [6] | M. J. Greenberg, Lectures on forms in many variables, New York-Amsterdam 1969. · Zbl 0185.08304 |
| [7] | J. Algebra 174 pp 1091– (1995) |
| [8] | Lewis D. J., Monatsh. Math. 97 pp 277– (1984) |
| [9] | Notices Amer. Math. Soc. 46 pp 2– (1999) |
| [10] | S., Trans. Amer. Math. Soc. 352 pp 5045– (2000) |
| [11] | R., Acta Math. 162 pp 1– (1989) |
| [12] | R. C. Vaughan, The Hardy-Littlewood method, second edition, Cambridge University Press, Cambridge1997. · Zbl 0868.11046 |
| [13] | T., Math. 419 pp 141– (1991) |
| [14] | Wooley T. D., Ann. Math. 135 pp 131– (1992) |
| [15] | T., London Math. Soc. Lect. Note Ser. 247 pp 361– (1997) |
| [16] | T., Bull. London Math. Soc. 29 pp 556– (1997) |
| [17] | T., Math. 488 pp 79– (1997) |
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