The digits of n+t – Lukas Spiegelhofer

Lukas Spiegelhofer
TU Vienna

Date(s) : 29/09/2020
11 h 15 min - 12 h 15 min

We study the binary sum-of-digits function s₂ under addition of a constant t∈ℕ.
For each integer j, we are interested in the asymptotic density
δ(j, t) = dens{n∈ℕ : s₂(n+t) – s₂(n) = j}.
In this talk, we consider the following two questions.
(1) Do we have
ct = δ(0, t) + δ(1, t) + … > 1/2 ?
This is a conjecture due to T. W. Cusick (2011).
(2) What does the probability distribution defined by j ↦ δ(j, t) look like?

We prove that indeed ct > 1/2 if the binary expansion of t contains at least M₀ blocks of contiguous 1’s, where M₀ is an absolute, effective constant.
Note that the number of exceptional t<T can easily be bounded by some power of log T.

Our second theorem states that δ(j, t) usually behaves like a normal distribution.
If M is the number of blocks of 1’s in t, where M ≥ M₀, we have
δ(j, t) = 1/√(2πv) exp( -j²/(2v) ) + O(M⁻¹(log M)⁴),
uniformly for j∈ℤ and with an absolute implied constant.
Here the variance v depends (explicitly) on the binary expansion of t.

This is joint work with Michael Wallner (TU Wien).

CIRM, Luminy


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