What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension?

Let 1 < s < 2, and λk > 0 with λk → ∞ satisfy the Hadamard condition λk+1/λk ⩾ λ > 1. For a class of Besicovich functions B(t)=∑k=1∞λks-2sin(λkt), the present paper investigates the intrinsic relationship between box dimension of graphs of their vth fractional integrals g(t) and uth fractional derivatives g˜(t) and the asymptotic behavior of {λk}. We show that: if 0 < v < 1, s > 1 + v, then for sufficiently large λ, dim¯BΓ(g)=dim̲BΓ(g)=s-v holds if and only if limn→∞logλn+1logλn=1; if 0 < u < 2 − s, then for sufficiently large λ, dim¯BΓ(g˜)=dim̲BΓ(g˜)=s+u holds if and only if limn→∞logλn+1logλn=1.