Consider an exponential polynomial $$f(t)=\sum_{k=1}^na_k\exp(i\lambda_kt),$$ where $a_k$ are complex and $\lambda_k, t$ real. The usual form of the Mean Motion Theorem says that the limit $$\lim_{t\to+\infty}\frac{\arg f(t)}{t}$$ exists. (If $f$ has real zeros one defines $\arg f$ by bypassing them along small half-circles in the upper half-plane).
All books that I know mention that this was conjectured by Lagrange, and proved by P. Bohl for $n=3$ and by B. Jessen and H. Tornehave (1945) in general, after earlier incomplete proofs by H. Weyl and P. Hartman. However in the paper of Bohl, a much more subtle question is actually studied, namely whether we have $$\arg f(t)=ct+O(1).$$ He shows that for $n=3$ this is sometimes the case, sometimes not, and gives an exact condition in terms of $a_k,\lambda_k$. My question is:
Has anyone ever continued this line of inquiery? Can $$\arg f(t)=ct+o(t)$$ be improved: a) in general, b) under some additional conditions on $a_k,\lambda_k$?
Of course one such condition is known since Lagrange: if $|a_1|>\sum_{k=2}^n|a_k|$, then the error term is $O(1)$.
Ref. P. Bohl, "Über ein in der Theorie der säkularen Störungen vorkommendes Problem", J. reine angew Math. 135 (1909) 189-283. There is a Russian translation in P. Bohl, Collected Works, Riga, Znanie, 1974.
