Despite the combination of math, a character named "Hilbert", and a hotel, this is suprisingly not a work of fiction that uses the analogy of an infinite hotel to explore the odd properties of countable infinities. (See here and here for some that do.)

That is not to say that the hotel in this story isn't infinite in size. It arguably is! And there are two passing remarks in the book which comment on the consequences of that.

However, the fact that the hotel is infinite is not nearly as important to the story as that one can walk through its corridors and exit in different cities (on different continents), that the layout is constantly evolving, and that the hotel itself is sentient.

Here are some observations and remarks, only some of which are mathematical in nature:

• A major focus of the book is on Neil's unusual job: he attends conferences on behalf of paying customers who don't want to attend themselves. The first half of the book mostly uses this to parody what happens at business conventions.
• The book has a psychodelic vibe, and is as much fantasy as it is science fiction. There are some bizarrely cool descriptions in the book that I personally enjoyed. The fact that the hallways of the hotel are never quite the same resonates with the way I myself get lost on trips that include stays at different hotels. ("Wasn't the elevator to the left yesterday?!?")
• Dee, a secondary protagonist, is an employee of the Way Inn. Her job is to find new locations for the hotel, which we at first think means that she chooses a location where construction workers will build an entirely new structure. Only later does it become clear that she helps the single infinite hotel structure grow through some interdimensional space to reach a new place on Earth.
• Dee explains to Neil that there is a pattern to the abstract paintings in the hotel rooms, that they are not separate pieces art but together convey some hidden message. (In the book this is described in only vaguely mathematical terms, such as "a representation of a spatial relationship".)
• Aside from the name, the character named "Hilbert" has no similarities to mathematician David Hilbert as far as I can tell.
• Neil realizes that the hotel "goes on forever" after Dee takes him on a tour which includes stops in different time zones, though he really has no way to know whether it is "very, very big" rather than "literally infinite in size" as far as I can tell. Neil soon asks about the light bulbs: "Who changes them? The number that need changing every day must exceed all the resources of the universe." Much later, when trying to find someone to help him, he notes "Of course. If the hotel was infinite and the number of guests remained finite, the chances of coming across an occupied room would be infinitesimally small." Near the end he describes the hotel as a "fractal continuity".
• Dee's speculations about the hotel include "A divergent pseudostructure. A non-Euclidean manifold. A prism projecting a hypersurface onto our space time from a point...a point outside."
• When Neil asks Dee how she is able to change the shape of the hotel, her reply is "Maths and meditation". Later she adds "That's on top of years of study of topography and topology at the doctoral level, and a gift for pattern recognition."
• When Neil returns from a failed escape attempt only to see himself running off in the distance at the start of the attempt, he thinks about "strange loops" and "Möbius strips".
• Dee's hotel room looks like the messy office of many a stereotypical mathematician in fiction, covered in papers and "geometric doodles".

This book was brought to my attention by a site visitor using "prospectscot" as an e-mail address. I don't know who that is, but I am grateful for the recommendation. Thanks!