Monatshefte für Mathematik

, Volume 188, Issue 1, pp 31–35

# Almost equal summands in Waring’s problem with shifts

• Kirsti D. Biggs
Open Access
Article

## Abstract

A result of Wright from 1937 shows that there are arbitrarily large natural numbers which cannot be represented as sums of s kth 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 11P05
Waring’s problem with shifts asks whether, given $$k,s\in \mathbb {N}$$ and $$\eta \in (0,1]$$, along with shifts $$\theta _1,\cdots ,\theta _s\in (0,1)$$ with $$\theta _1\not \in \mathbb {Q}$$, we can find solutions in natural numbers $$x_i$$ to the following inequality, for all sufficiently large $$\tau \in \mathbb {R}$$:
\begin{aligned} \left| (x_1-\theta _1)^k+\cdots +(x_s-\theta _s)^k-\tau \right| <\eta . \end{aligned}
(1)
This problem was originally studied by Chow in [3]. In [1], the author showed that an asymptotic formula for the number of solutions to (1) can be obtained whenever $$k\ge 4$$ and $$s\ge k^2+(3k-1)/4$$. The corresponding result for $$k=3$$ and $$s\ge 11$$ is due to Chow in [2].
An interesting variant is to consider solutions of (1) subject to the additional condition
\begin{aligned} \left| x_i-(\tau /s)^{1/k}\right| <y(\tau ),\quad (1\le i\le s), \end{aligned}
for some function $$y(\tau )$$. In other words, we are confining our variables to be within a small distance of the “average” value.

In 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 kth 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 $$n\rightarrow \infty$$.

In [4] and [5], Daemen showed that if we widen the permitted region slightly, we can once again guarantee solutions in the classical case. Specifically, he obtains a lower bound on the number of solutions under the condition
\begin{aligned} \left| x_i-(n/s)^{1/k}\right| <cn^{1/2k},\quad (1\le i\le s), \end{aligned}
for a suitably large constant c, and an asymptotic formula under the condition
\begin{aligned} \left| x_i-(n/s)^{1/k}\right| <n^{1/2k+\epsilon },\quad (1\le i\le s). \end{aligned}
In this note, we show that (a slight strengthening of) Wright’s result remains true in the shifted case. Specifically, we prove the following.

### 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 sk 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 $$\eta \ll m^{k-2}$$.

Suppose $$\tau _m$$ satisfies (1) with $$0<\eta <c\tau _m^{1-2/k}$$ and $$\left| x_i-(\tau _m/s)^{1/k}\right| <c'\tau _m^{1/2k}$$ for $$1\le i\le s$$. We write $$x_i=m+a_i$$, and observe that
\begin{aligned} m^{k-1}\left| a_i\right|&=m^{k-1}\left| x_i-m\right| \\&\le m^{k-1}\Big (\left| x_i-(\tau _m/s)^{1/k}\right| +\left| (\tau _m/s)^{1/k}-m\right| \Big )\\&\le c'm^{k-1}\tau _m^{1/2k}+\left| \tau _m/s-m^k\right| . \end{aligned}
Using the definition of $$\tau _m$$, we obtain
\begin{aligned} m^{k-1}\left| a_i\right|&\le c_1m^{k-1}m^{1/2}+km^{k-1}\left( 1-s^{-1}\sum _{i=1}^s \theta _i\right) , \end{aligned}
and therefore $$\left| a_i\right| \le c_2 m^{1/2}$$ for $$1\le i\le s$$. Expanding (1), we see that
\begin{aligned} \eta&> \left| \sum _{i=1}^s (x_i-\theta _i)^k - \tau _m\right| \nonumber \\&=\left| \sum _{i=1}^s (m+a_i-\theta _i)^k - \Big (sm^{k}+km^{k-1}(s-\sum _{i=1}^s \theta _i)\Big )\right| \\&\ge km^{k-1}\left| s-\sum _{i=1}^s a_i \right| -\left| \sum _{j=2}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) m^{k-j}\sum _{i=1}^s(a_i-\theta _i)^j\right| .\nonumber \end{aligned}
(2)
Rearranging, this gives
\begin{aligned} \left| s-\sum _{i=1}^s a_i \right|&<\eta k^{-1}m^{1-k} +\left| \sum _{j=2}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) k^{-1}m^{1-j} \sum _{i=1}^s(a_i-\theta _i)^j\right| \\&\le \eta k^{-1}m^{1-k}+\sum _{j=2}^k \left( {\begin{array}{c}k\\ j\end{array}}\right) k^{-1}m^{1-j} s(c_3 m^{1/2})^j\\&\le c_4. \end{aligned}
By choosing our original $$c,c'$$ to be sufficiently small, we may conclude that $$c_4\le 1$$, which implies that $$\sum _{i=1}^s a_i = s$$. Substituting this back into (2), when $$k=2$$ we obtain
\begin{aligned} \eta&>\left( {\begin{array}{c}k\\ 2\end{array}}\right) m^{k-2}\sum _{i=1}^s(a_i-\theta _i)^2, \end{aligned}
and consequently
\begin{aligned} \sum _{i=1}^s(a_i-\theta _i)^2 < c_5, \end{aligned}
which is a contradiction if we choose $$c,c'$$ sufficiently small, since we know that $$\sum _{i=1}^s(a_i-\theta _i)^2\gg 1$$.
When $$k\ge 3$$, we obtain
\begin{aligned} \eta&>\left| \sum _{j=2}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) m^{k-j}\sum _{i=1}^s (a_i-\theta _i)^j\right| \\&\ge \left( {\begin{array}{c}k\\ 2\end{array}}\right) m^{k-2}\sum _{i=1}^s(a_i-\theta _i)^2 -\left| \sum _{j=3}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) m^{k-j}\sum _{i=1}^s (a_i-\theta _i)^j\right| . \end{aligned}
Consequently,
\begin{aligned} \left( {\begin{array}{c}k\\ 2\end{array}}\right) m^{k-2}\sum _{i=1}^s(a_i-\theta _i)^2&<\eta +\sum _{j=3}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) m^{k-j}\sum _{i=1}^s \left| a_i-\theta _i\right| ^j\\&\le \eta +\sum _{j=3}^k\left( {\begin{array}{c}k\\ j\end{array}}\right) m^{k-j}(c_3 m^{1/2})^{j-2} \sum _{i=1}^s (a_i-\theta _i)^2\\&\le \eta +c_6 m^{k-5/2}\sum _{i=1}^s (a_i-\theta _i)^2, \end{aligned}
and so
\begin{aligned} \sum _{i=1}^s(a_i-\theta _i)^2&<c_7+c_8 m^{-1/2}\sum _{i=1}^s(a_i-\theta _i)^2, \end{aligned}
which is again a contradiction when 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

For $$s,k\ge 2$$ natural numbers, $$\varvec{\theta }=(\theta _1,\cdots ,\theta _s)\in (0,1)^s$$, and suitably small constants $$C,C'>0$$, there exist arbitrarily wide gaps between real numbers $$\tau$$ for which the system
\begin{aligned} \begin{aligned}&\left| (x_1-\theta _1)^k+\cdots +(x_s-\theta _s)^k -\tau \right|<C\tau ^{1-2/k}\\&\left| x_i-(\tau /s)^{1/k}\right| <C'\tau ^{1/2k},\quad (1\le i\le s) \end{aligned} \end{aligned}
(3)
has a solution in natural numbers $$x_1,\cdots ,x_s$$.

### 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.

We have
\begin{aligned} \left| (\tau /s)^{1/k}-(\tau _0/s)^{1/k}\right|&\le s^{-1/k}\left| (\tau _0-\delta )^{1/k}-\tau _0^{1/k}\right| \\&\le C_1 \delta \tau _0^{1/k-1}, \end{aligned}
and consequently
\begin{aligned} \left| x_i-(\tau _0/s)^{1/k}\right|&\le \left| x_i-(\tau /s)^{1/k} \right| +\left| (\tau /s)^{1/k}-(\tau _0/s)^{1/k}\right| \\&< C'\tau ^{1/2k}+C_1 \delta \tau _0^{1/k-1}\\&\le C'(\tau _0+\delta )^{1/2k}+ C_1C_0\tau _0^{-1/k}\\&\le C_2 \tau _0^{1/2k}. \end{aligned}
We also see that
\begin{aligned} \left| \sum _{i=1}^s (x_i-\theta _i)^k-\tau _0\right|&\le \left| \sum _{i=1}^s (x_i-\theta _i)^k-\tau \right| +\left| \tau -\tau _0\right| \\&< C\tau ^{1-2/k}+\delta \\&\le C(\tau _0+\delta )^{1-2/k}+C_0\tau _0^{1-2/k}\\&\le C_3 \tau _0^{1-2/k}. \end{aligned}
Choosing $$C_0,C,C'$$ small enough to ensure that $$C_2\le c'$$ and $$C_3\le c$$ gives a contradiction to our original choice of $$\tau _0$$. Consequently, there is no solution to (3) in an interval of radius $$\asymp \tau _0^{1-2/k}$$ around $$\tau _0$$. $$\square$$

## Notes

### Acknowledgements

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

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