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APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV’S MEAN VALUE THEOREM

Part of: Exponential sums and character sums Additive number theory; partitions

Published online by Cambridge University Press:  21 December 2016

Trevor D. Wooley
Trevor D. Wooley*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, U.K. email [email protected]
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Abstract

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$.

MSC classification

Primary: 11L15: Weyl sums 11L07: Estimates on exponential sums 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 292 - 350
Copyright
Copyright © University College London 2016 

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References

Arkhipov, G. I., Chubarikov, V. N. and Karatsuba, A. A., Trigonometric Sums in Number Theory and Analysis (de Gruyter Expositions in Mathematics 39 ), Walter de Gruyter (Berlin, 2004).Google Scholar
Bourgain, J., Demeter, C. and Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2) 184 2016, 633–682.Google Scholar
Ford, K. B., New estimates for mean values of Weyl sums. Int. Math. Res. Not. IMRN 1995(3) 1995, 155–171.CrossRefGoogle Scholar
Ford, K. B. and Wooley, T. D., On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing. Acta Math. 213(2) 2014, 199–236.Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 4th edn., The Clarendon Press, Oxford University Press (Oxford, 1960).Google Scholar
Hua, L.-K., On Tarry’s problem. Q. J. Math. Oxford 9 1938, 315–320.Google Scholar
Hua, L.-K., Improvement of a result of Wright. J. Lond. Math. Soc. (2) 24 1949, 157–159.CrossRefGoogle Scholar
Hua, L.-K., Additive Theory of Prime Numbers, American Mathematical Society (Providence, RI, 1965).Google Scholar
Vaughan, R. C., The Hardy–Littlewood Method, Cambridge University Press (Cambridge, 1997).Google Scholar
Vaughan, R. C. and Wooley, T. D., A special case of Vinogradov’s mean value theorem. Acta Arith. 79(3) 1997, 193–204.CrossRefGoogle Scholar
Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers. Trav. Inst. Math. Stekloff 23 1947.Google Scholar
Wooley, T. D., On Vinogradov’s mean value theorem. Mathematika 39(2) 1992, 379–399.CrossRefGoogle Scholar
Wooley, T. D., A note on simultaneous congruences. J. Number Theory 58(2) 1996, 288–297.Google Scholar
Wooley, T. D., Some remarks on Vinogradov’s mean value theorem and Tarry’s problem. Monatsh. Math. 122(3) 1996, 265–273.CrossRefGoogle Scholar
Wooley, T. D., Vinogradov’s mean value theorem via efficient congruencing. Ann. of Math. (2) 175(3) 2012, 1575–1627.Google Scholar
Wooley, T. D., The asymptotic formula in Waring’s problem. Int. Math. Res. Not. IMRN 2012(7) 2012, 1485–1504.CrossRefGoogle Scholar
Wooley, T. D., Vinogradov’s mean value theorem via efficient congruencing, II. Duke Math. J. 162(4) 2013, 673–730.Google Scholar
Wooley, T. D., Translation invariance, exponential sums, and Waring’s problem. In Proc. Int. Congr. Mathematicians (Seoul, Korea, 13–21 August 2014), Vol. II, Kyung Moon Sa (Seoul, 2014), 505–529.Google Scholar
Wooley, T. D., Multigrade efficient congruencing and Vinogradov’s mean value theorem. Proc. Lond. Math. Soc. (3) 111(3) 2015, 519–560.CrossRefGoogle Scholar
Wooley, T. D., The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294 2016, 532–561.CrossRefGoogle Scholar
Wright, E. M., The Prouhet–Lehmer problem. J. Lond. Math. Soc. (2) 23 1948, 279–285.Google Scholar