Small values of signed harmonic sums

Abstract

For every τ ∈ R and every integer N , let mN (τ ) be the minimum of the distance of τ from the sums N n=1 sn/n, where s1, . . . , sn ∈ {−1, +1}. We prove that mN (τ ) < exp − C(log N ) 2 , for all sufficiently large positive integers N (depending on C and τ ), where C is any positive constant less than 1/ log 4.

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

“…Indeed, one obtains that, for almost all τ , the distance |τ − σ n (τ )| is infinitely often as small as e −(log n) 2 / log 4(1+o (1)) and that this bound is optimal. It is quite remarkable that the sequence s n (τ ) allows one to recover, albeit for a subsequence and only for almost all τ , exactly the estimate (1.1) obtainable with probabilistic methods [11].…”

“…A natural question is then to determine whether one can find sequences (s n ) n∈N such that the partial sums σ n := n This approach was considered by the authors [11], where it is shown that…”

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 , . .…”