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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.

References

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Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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