Waring's problem with shifts

Kirsti Biggs (Bristol)

Frank Adams 1,

In its original form, Waring's problem asks whether every positive integer can be written as the sum of \(s\) \(k\)th powers of natural numbers, where \(s\) depends only on \(k\). In this talk, I will discuss an analogue of this problem in which we attempt to approximate a large, positive real number \(\tau\) by a sum of `shifted' \(k\)th powers. I will outline the Davenport--Heilbronn method, which allows us to obtain an asymptotic formula for the number of solutions to the relevant Diophantine inequality whenever \(s\geq k^2+(3k-1)/4\), which improves on the best previously known result. I will also show that there are arbitrarily large \(\tau\) which cannot be approximated in this way if we insist on the variables \(x_i\) being too close together.
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