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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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 \(\tau \in \mathbb {R}\):
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
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
for a suitably large constant c, and an asymptotic formula under the condition
In this note, we show that (a slight strengthening of) Wright’s result remains true in the shifted case. Specifically, we prove the following.
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 ProofThis 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
Using the definition of \(\tau _m\), we obtain
and therefore \(\left| a_i\right| \le c_2 m^{1/2}\) for \(1\le i\le s\). Expanding (1), we see that
Rearranging, this gives
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
and consequently
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
Consequently,
and so
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 2For \(s,k\ge 2\) 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
has a solution in natural numbers \(x_1,\cdots ,x_s\).
FormalPara ProofBy 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
and consequently
We also see that
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 \(\tau _0\). \(\square \)
References
Biggs, K.D.: On the asymptotic formula in Waring’s problem with shifts. J. Number. Theory (2018). https://doi.org/10.1016/j.jnt.2017.12.009
Chow, S.: Sums of cubes with shifts. J. Lond. Math. Soc. (2) 91(2), 343–366 (2015)
Chow, S.: Waring’s problem with shifts. Mathematika 62(1), 13–46 (2016)
Daemen, D.: The asymptotic formula for localized solutions in Waring’s problem and approximations to Weyl sums. Bull. Lond. Math. Soc. 42(1), 75–82 (2010)
Daemen, D.: Localized solutions in Waring’s problem: the lower bound. Acta Arith. 142(2), 129–143 (2010)
Wright, E.M.: The representation of a number as a sum of four ‘almost equal’ squares. Q. J. Math. 8, 278–279 (1937)
Acknowledgements
The author would like to thank Trevor Wooley for his supervision, and the anonymous referee for useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
The author is supported by EPSRC Doctoral Training Partnership EP/M507994/1.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Biggs, K.D. Almost equal summands in Waring’s problem with shifts. Monatsh Math 188, 31–35 (2019). https://doi.org/10.1007/s00605-018-1178-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-018-1178-7