A Hilbert cube of dimension d in the integers is a set of the form {a + Σ ε_i b_i : ε_i ∈ {0,1}} for distinct positive integers b_1, ..., b_d — a combinatorial hypercube embedded in arithmetic. The question is: how large can a Hilbert cube be inside a set with specific arithmetic structure?
Croot, Mao, and Yip (arXiv:2603.14654) develop general frameworks for bounding the maximal dimension of Hilbert cubes inside sets defined by arithmetic properties — perfect powers, primes, smooth numbers. Their results, conditional on the ABC conjecture, yield nearly sharp uniform upper bounds on the number of k-th powers in an arithmetic progression for k ≥ 4.
The ABC conjecture enters because it controls how frequently integers with structured prime factorizations can be additively related. A Hilbert cube inside the perfect k-th powers means that many specific sums and differences of k-th powers are themselves k-th powers. The ABC conjecture limits how often this can happen by bounding the radical (product of distinct prime factors) of integers involved in additive relations.
The bound is almost sharp: for k ≥ 4, the maximal number of k-th powers in an arithmetic progression of length N is bounded by a function that matches known constructions up to logarithmic factors. The gap between upper and lower bounds is narrow enough to suggest that the true answer is close to being determined.
The method extends beyond perfect powers. Smooth numbers (integers with only small prime factors), primes, and other arithmetic sets have different Hilbert cube capacities, each bounded by the framework. The unifying idea is that arithmetic structure in a set constrains its combinatorial structure — sets defined by multiplicative properties cannot contain large additive hypercubes.