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.

FormalPara 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 \(1\le i\le s\).

FormalPara 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. \(\square \)

FormalPara 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 \(x_1,\cdots ,x_s\).

FormalPara 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 \(\tau _0\). \(\square \)