Solid state physics problems of that sort are used in the process of engineering new materials (e.g. a more efficient dielectric for capacitors, a magnetic material that can store magnetic memory data at higher temperatures without destablizing, a higher temperature superconductor, etc). Idk if this particular result has such an application, but I wouldn't be surprised if it did.
In addition to solid state physics, "special functions" show up in many areas of applied math and physics, including the design, calibration, and data analysis of/from MRI machines and antennas. Many special functions including the hypergeometric functions can be approximated more efficiently using fancy number theory algorithms.
There is also a broad category of physics and chemistry problems that are best solved by Monte Carlo simulations, and sometimes Monte Carlo approximations can be made using many fewer iterations by choosing a set of points within the sample space that satisfy number theoretic constraints.
Also, physics models often inspire machine learning models so it's not impossible that some niche ML algorithm could eventually benefit from a faster algorithm that uses this result or a derivative of it.
None of these directly answer your question about a definitive application for this particular number theory result, but I hope it gives you an idea of what it could potentially be used for. The closest analogy I know if is that certain multi-dimensional Monte Carlo integral approximations can be done orders of magnitude more quickly if you choose lattice points whose coordinates satisfy some similar-ish constraints to the ones described for this problem.