Georgia Institute of Technology College of Sciences

A unit fraction representation of a positive rational number writes it as a sum of finitely many unit fractions $\frac{1}{n}$ with distinct positive integers $n$. In 1202, Fibonacci showed that every positive rational number has such a representation. Since then, people have been interested in representations with special properties. In this spirit, Bloom asked: which sets $A \subseteq \mathbb{N}$ can represent every positive rational number, subject to some local conditions, using only denominators from $A$? Croot proved in 2003 that for any finite colouring of $\mathbb{N}$, $1$ can be written as sum of unit fractions with denominators in a single-color class. Bloom showed that any $A$ with positive upper logarithmic density works, and Liu and Sawhney improved this to sets with upper logarithmic density $(\log N)^{-1/5 + o(1)}$. Bloom also showed that the shifted primes $p-1$ have this property.

To characterize all sets $A$ with this special property, we need to understand which arithmetic properties make this possible. One way to do that is to study thinner sets that still have this special property. In this talk we show that the shifted Golomb primes have this property. The Golomb primes are a subset of primes defined by $q_1 = 3$ and, once $q_1, \dots, q_k$ are known, $q_{k+1}$ is the smallest prime $p$ with $p \not\equiv 1 \pmod{q_i}$ for all $i \le k$. They are a natural generalization of the primes, and their relative density in the primes is about $1/(loglogx)$, which goes to $0$.