Abstract. We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.
We apply a recent refinement of the Hardy-Littlewood method to obtain an asymptotic lower bound for the number of solutions of a linear diophantine inequality in three prime variables. Using the same ideas, we are able to show that a linear form in two primes closely approximates almost all real numbers lying in a suitably discrete set. # 2002 Elsevier Science (USA)
Write APY R fn e 1Y P Z X pjn A p Rg, and de®ne the exponential sum f Y PY R xY y e APY R J 2r zY wY c Q 0 and J 2r zYwY c Q 0Y 3X6 where (recalling the notation of the previous section) we have putParsell, Multiple exponential sums over smooth numberswhere H r JÀ1 a2r J . The second term here isParsell, Multiple exponential sums over smooth numbers
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