Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

Question

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere?

I think the answer is yes but I am not sure how to prove it.

If we allow any square of any size the answer is yes - an old Putnam problem. You obtain the same result if you restrict to squares of any size parallel to the axes. However, if you take only unit squares parallel to the axes the answer is no.

Note that an obvious idea for a counterexample is to try to colour the plane with 2 colours so that any two points a unit distance apart have different colours. However this is trivial to disprove.

The title and the body are asking in the opposite senses, so "yes" could mean "yes, there is a nonzero function ..." or "yes, a function ... has to vanish". — Jukka Kohonen
– Jukka Kohonen, Commented Mar 7, 2024 at 9:38
Where might one find a proof of the result concerning squares of any size parallel to the axes? — Gro-Tsen
– Gro-Tsen, Commented Mar 7, 2024 at 18:12
@Gro-Tsen I might have messed up this calculation, but I think that if you let $S$ be the set of integer points $(i,j)$ with $1\le i\le 7$ and $1\le j \le 7$, and write down the $1^2 + 2^2 + \cdots + 6^2$ linear equations implied by the condition, then you get a matrix of full rank. — Timothy Chow
– Timothy Chow, Commented Mar 7, 2024 at 22:58
@Gro-Tsen You can actually prove this for $\mathbb{Z}^2$. Denote the vertex values in any 2x2 square by \begin{bmatrix} a & b & c\\ d & e & f\\ g &h & i \end{bmatrix} Using the 1x1 square sums we add $a+b+d+e=0$ to $e+f+h+i=0$ and subtract $d+e+g+h=0$ and $b+c+e+f=0$ giving $a+i=g+c$. The 2x2 corner square sum is $a+c+i+g=0$ so $a+i=g+c=0$. Hence any 2x2 square has corner values of the form \begin{bmatrix} a & -g\\ g & -a \end{bmatrix} This implies any 4x4 square has corner values all equal: \begin{bmatrix} a & a\\ a & a \end{bmatrix} implying $a=0$. — Ivan Meir
– Ivan Meir, Commented Mar 8, 2024 at 2:00

Emil Jeřábek · Accepted Answer · 2024-03-06 13:56:48Z

28

The answer is yes, I learned this from the paper that appeared on arXiv literally yesterday https://arxiv.org/abs/2403.01279, they quote

R. Katz, M. Krebs and A. Shaheen, Zero sums on unit square vertex sets and plane colorings, Amer. Math. Monthly 121 (2014), no. 7, 610–618.

for this exact result about the unit square.

edited Mar 6, 2024 at 13:56

Emil Jeřábek

50.6k4 gold badges162 silver badges224 bronze badges

answered Mar 6, 2024 at 13:46

Aleksei Kulikov

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Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

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