# Almost equal summands in Waring’s problem with shifts

## Abstract

A result of Wright from 1937 shows that there are arbitrarily large natural numbers which cannot be represented as sums of *s* *k*th powers of natural numbers which are constrained to lie within a narrow region. We show that the analogue of this result holds in the shifted version of Waring’s problem.

## Keywords

Waring’s problem Diophantine inequalities Shifted integers## Mathematics Subject Classification

11D75 11P05In 1937, Wright studied this question in the setting of the classical version of Waring’s problem, and proved in [6] that there exist arbitrarily large natural numbers *n* which cannot be represented as sums of *s* *k*th powers of natural numbers \(x_i\) satisfying the condition \(\left| x_i^k-n/s\right| <n^{1-1/2k}\phi (n)\) for \(1\le i\le s\), no matter how large *s* is taken. Here, \(\phi (n)\) is a function satisfying \(\phi (n)\rightarrow 0\) as \(n\rightarrow \infty \).

*c*, and an asymptotic formula under the condition

### Theorem 1

Let \(s,k\ge 2\) be natural numbers. Fix \(\varvec{\theta }=(\theta _1,\cdots ,\theta _s)\in (0,1)^s\), and let \(c,c'>0\) be suitably small constants which may depend on *s*, *k* and \(\varvec{\theta }\). There exist arbitrarily large values of \(\tau \in \mathbb {R}\) which cannot be approximated in the form (1), with \(0<\eta <c\tau ^{1-2/k}\), subject to the additional condition that \(\left| x_i-(\tau /s)^{1/k}\right| <c'\tau ^{1/2k}\) for \(1\le i\le s\).

### Proof

This follows the structure of Wright’s proof in [6], with minor adjustments to take into account the shifts present in our problem. As such, for \(m\in \mathbb {N}\), we let \(\tau _m=sm^{k}+km^{k-1}(s-\sum _{i=1}^s \theta _i)\), and we note that \(\tau _m\rightarrow \infty \) as \(m\rightarrow \infty \). Throughout the proof, we allow \(c_1,c_2,\cdots \) to denote positive constants which do not depend on *m*, although they may depend on the fixed values of \(s,k, \varvec{\theta }, c\) and \(c'\). We also note that \(\eta <c\tau ^{1-2/k}\) implies that \(\eta \ll m^{k-2}\).

*m*is large.

We conclude that for all sufficiently large *m*, it is impossible to approximate \(\tau _m\) in the manner claimed. This completes the proof. \(\square \)

### Corollary 2

### Proof

By Theorem 1, we fix \(\tau _0\in \mathbb {R}\) such that there is no solution in natural numbers \(x_1,\cdots ,x_s\) to \(\left| (x_1-\theta _1)^k+\cdots +(x_s-\theta _s)^k-\tau _0\right| <c\tau _0^{1-2/k}\) with \(\left| x_i-(\tau _0/s)^{1/k}\right| <c'\tau _0^{1/2k}\) for \(1\le i\le s\).

Let \(0<\delta \le C_0\tau _0^{1-2/k}\) for some \(C_0>0\), and let \(\tau \in [\tau _0-\delta ,\tau _0+\delta ]\). Let \(C,C'>0\) be suitably small constants depending on \(c,c'\) and \(C_0\) to be chosen later, and suppose that \(x_1\cdots ,x_s\in \mathbb {N}\) are such that (3) is satisfied.

## Notes

### Acknowledgements

The author would like to thank Trevor Wooley for his supervision, and the anonymous referee for useful comments.

## References

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