Greedy approximations by signed harmonic sums and the Thue–Morse sequence

Abstract

Given a real number τ , we study the approximation of τ by signed harmonic sums σN (τ ) := n≤N sn(τ )/n, where the sequence of signs (sN (τ )) N ∈N is defined "greedily" by setting sN+1(τ ) := +1 if σN (τ ) ≤ τ , and sN+1(τ ) := −1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence (σN (τ ) − τ ) N ∈N . Moreover, we give an accurate description of the behavior of the sequence of signs (sN (τ )) N ∈N , highlighting a surprising connection with the Thue-Morse sequence.

Infinite products related to generalized Thue-Morse sequences

“…In order to prove that (3.4) converges, we only need to show that ∞ n=1 δ n n converges. Enlightened by partial summation (see for example the equality (6.5) in [8] related to the Thue-Morse sequence), we consider the following 1 and 2 , which complete the proof. 1 Prove that…”

Small values of signed harmonic sums

“…We are assuming that log N ≤ log x ≤ C(log N ) 2 , hence this is implied by C/C log N + log log N ≤ log N + O C,C (1) which is true as soon as N is large enough. This proves that we can apply Lemma 3.9, getting δ 2 · #S 1 (N, δ, | N (x)| ≤ exp − π 2 δ 2 2 · #S 1 (N, δ, x) ≤ exp − π 2 400 2 exp(E log x) , which is the claim.…”

“…for infinitely many N (see [1,Proposition 5.9]). The bound in (1.3) is not optimal, and some minor variations of our proof are already able to produce some small improvement.…”

“…The study of this "greedy" sequences needs completely different tools from those employed here, thus we leave its study to another paper [1].…”

Counting Egyptian fractions

“…as N → +∞. Another motivation for studying the cardinality of E N comes from the recent work of three of the authors [2] (see also [1]), where the question of how well a real number τ can be approximated by sums of the form N n=1 s n /n, where s 1 , . .…”