Motivated by a problem concerning the existence of quasi-periodic solutions in dynamical systems, I encountered a number-theoretic question that appears somewhat unusual. I would like to know whether there exists an infinite sequence of positive odd integers ${j_1, j_2, \ldots, j_k, \ldots}$ satisfying the following conditions:
- The sequence is strictly increasing: $j_1 < j_2 < \cdots$;
- The fractional parts $\{j_k\sqrt{2}\}$ lie in the interval $[1/4, 3/4]$ for all $k \geq 1$;
- For all indices $k, l, m$ with $k,l<m$, and for all factorizations of the form $j_m^2 -j_l^2=ab$ and $j_m^2 -j_k^2=cd$ where $a, b, c, d$ are positive integers with $a > 2 j_m$ and $c > 2 j_m$, the relation $a=b+c+d$ never holds.
The existence of such a sequence seems plausible, as the condition $a = b + c + d$ appears to be highly restrictive and may occur only coincidentally. Numerical experiments in Mathematica support this intuition: I was able to construct a finite sequence of length over 40 satisfying the above properties. For instance, one such sequence is:
{9,11,13,21,23,49,57,83,185,243,257,539,561,585,989,1165,1203,1397,1561,1623,2297,2341,2449,2609,
2725,3043,3231,3331,3429,3787,4371,5571,6481,6587,6857,6973,7377,8057,8495,8577,8671,9895}.
I would greatly appreciate any insights or references regarding the existence or non-existence of such a sequence, or suggestions on how to approach this problem.
