Multigrade efficient congruencing and Vinogradov's mean value theorem Article Swipe
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Trevor D. Wooley
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YOU?
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· 2015
· Open Access
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· DOI: https://doi.org/10.1112/plms/pdv034
· OA: W1904038357
YOU?
·
· 2015
· Open Access
·
· DOI: https://doi.org/10.1112/plms/pdv034
· OA: W1904038357
We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.
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