3

This is Exercise 58 in Chapter one of our textbook Function Theory of One Complex Variable by Greene and Krantz:

Let $z_1, z_2, \ldots$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every $j,k$ (the integer may depend on $j$ and $k$), then prove that the $\{z_j\}$ lie on a single straight line.

I have no idea how to approach this problem. I tried to read the paper by Anning and Erd"os https://www.ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/S0002-9904-1945-08407-9.pdf, but I couldn't understand it and it does not use complex analysis.

Lingling
  • 105
  • 6
  • See this question – lulu Feb 03 '22 at 14:34
  • Thank you! I found in that question a follow up of the above mentioned paper by Erd"os https://www.ams.org/journals/bull/1945-51-12/S0002-9904-1945-08490-0/S0002-9904-1945-08490-0.pdf, which contains a shorter proof – Lingling Feb 03 '22 at 14:56
  • Is there any way to do it by complex analysis? – Lingling Feb 03 '22 at 14:57
  • Does the textbook imply that there is a complex analysis solution? Maybe it simply wants you to get used to metric geometry of the complex plane. What else is discussed in the chapter 1? – Moishe Kohan Feb 03 '22 at 17:39
  • Chapter 1 establishes the concept of complex differentiable, which includes Cauchy-Riemann equations, and also gives several direct properties of holomorphic functions and holomorphic polynomials. – Lingling Feb 03 '22 at 18:35
  • @Lingling Then, most likely, they do not expect you to use any complex analysis for this problem. – Moishe Kohan Feb 03 '22 at 18:58
  • Yes. Thank you! I think I'd better close this question on mathstack, but I don't know how to do it. – Lingling Feb 06 '22 at 14:39

0 Answers0