Binary versus Ternary

par Michael Drmota (TU Wien, Autriche)

lundi 18 nov. 2024, 11:00 → 12:00 Europe/Paris

Salle Grisvard, IHP, Paris

The relation between the binary and the ternary expansion of a given positive integer or a class of integers is still not completely understood. For example, we know almost nothing about the binary expansion of powers of 3. Only recently Spiegelhofer proved a "folklore conjecture" saying that there are infinitely many $n$ with $s_2(n)=s_3(n)$, where $s_2(n)$ and $s_3(n)$ denote the binary and ternary sum-of-digits functions, respectively.
The purpose of this talk is to present a far reaching generalization of this result. It is show that the set of pairs $(s_2(n),s_3(n))$ covers almost the whole first quadrant of lattice points (only with possible gaps in the boundary region). Interestingly the proof requires a combination of techniques from analytic number theory (Gowers norms, level-of-distribution results, exponential sums) and Diophantine approximation (Baker's theorem, $p$-adic subspace theorem).
This is joint work with Lukas Spiegelhofer.