# Controlling line thickness in ParametricPlot3D [duplicate]

I want to control the thickness of the line in ParametricPlot3D. I read the answer in
Thickness definition for ParametricPlot3D but it does not solve the problem. In the example below,

thickness = 3.0;ParametricPlot3D[{Sin[u], Cos[u], u/10}, {u, 0, 20}, PlotStyle -> AbsoluteThickness[thickness]]

the thickness changes abruptly from something very thin (for my taste) to something very thick (for my taste) by changing thickness from 3.0 to 3.1. Thickness seems to increase smoothly only above 3 or so. Is it possible to control the line thickness of thinner lines in a smooth way?

[Edited after the original posting] A new observation, when using \PlotStyle->Dashed the line thickness behaves as expected, it increases continuously from 0 up.