## Waring's problem with shifts

#### Kirsti Biggs (Bristol)

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.