On Vinogradov’s mean value theorem. (English) Zbl 0769.11036

This paper brings to the reader’s attention the important role played by iterative methods in the context of Vinogradov’s mean value theorem. Once this has been observed, it is possible to improve substantially upon the relevant estimates by adopting the “efficient differencing” introduced by the author [Ann. Math., II. Ser. 135, 131-164 (1992; Zbl 0754.11026)]. In the present context one is interested in upper bounds for the number \(J_{k,s}(P)\) of solutions to the system of diophantine equations \[ x^ l_ 1+\cdots+ x^ l_ s=y_ 1^ l+\cdots+ y^ l_ s\qquad (1\leq l\leq k) \] with \(1\leq x_ i\), \(y_ i\leq P\). The precise bounds obtained here are rather complicated to state and we refer to the paper for details. We mention two out of various applications: The asymptotic formula in Waring’s problem for \(k\)-th powers holds if the number of variables exceeds \(\bigl(2+o(1)\bigr)k^ 2 \log k\). This beats a result of Hua Lookeng (dating back to 1949) by a factor 2. Similarly, an asymptotic formula for \(J_{k,s}(P)\) holds when \(s\geq\bigl({5\over 3}+o(1)\bigr)k^ 3 \log k\). This improves the previous best by a factor 5/9.


11P05 Waring’s problem and variants
11D72 Diophantine equations in many variables


Zbl 0754.11026
Full Text: DOI


[1] DOI: 10.1093/qmath/os-9.1.199 · Zbl 0020.10504
[2] DOI: 10.1093/qmath/os-20.1.48 · Zbl 0039.27403
[3] DOI: 10.1007/BF01482074 · JFM 48.0146.01
[4] Baker, Diophantine Inequalities (1986)
[5] DOI: 10.2307/2946566 · Zbl 0754.11026
[6] Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie (1963)
[7] Vinogradov, Trav. Inst. Steklov 23 (1947)
[8] Vinogradov, Dokl. Akad. Nauk SSSR 8 pp 195– (1935)
[9] Vaughan, Mathematika 33 pp 6– (1986)
[10] Vaughan, J. Reine Angew. Math. 365 pp 122– (1986)
[11] Vaughan, The Hardy-Littlewood Method (1981)
[12] Stechkin, Trudy Mat. Inst. Steklov. 134 pp 283– (1975)
[13] Turina, Izv. Akad. Nauk SSSR, Ser. Mat. 51 pp 337– (1987)
[14] Linnik, Mat. Sbornik 12 pp 23– (1943)
[15] Karatsuba, Izv. Akad. Nauk SSSR 37 pp 1203– (1973)
[16] Hua, Additive Theory of Prime Numbers (1965) · Zbl 0192.39304
[17] DOI: 10.1112/jlms/s2-38.2.216 · Zbl 0619.10046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.