2020
DOI: 10.1016/j.aim.2020.107068
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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.

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Cited by 6 publications
(7 citation statements)
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“…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…”
Section: By I) and [6 Proposition 1] We Getmentioning
confidence: 92%
“…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…”
Section: By I) and [6 Proposition 1] We Getmentioning
confidence: 92%
“…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.…”
Section: 1mentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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“…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 , . .…”
Section: Introductionmentioning
confidence: 99%