The present paper discusses the relativistic Doppler effect and tries to found misunderstandings in the present state of the Special theory of relativity. The author's conclusion that he found some “blue shift” which contradicts with time dilation is wrong.
The weakest feature of the paper is that although the formulas, presented by authors, are in general correct, but they do not support the conclusions the author extract from them, and mistake is hidden in the interpretation.
Let's focus on the plane waves.
In general, the transverse Doppler effect, as it is studied in the available literature, means that an observer (let's call him the 1st observer), that receive an electromagnetic wave from a distant source, moving relative to the observer, will measure the wave frequency ν'=ν/γ, where γ=1/(sqrt(1-β²)), β²=u²/c², provided that the angle between the wave direction and the vector of the motion of the source, measured by the observer, is equal to π/2 (α'=π/2). So the light from moving source is red-shifted. It is generally treated as a pure effect of the special theory of relativity, and is due to the time dilation. Indeed, the observer can treat the wave crests as a clock, and the decrease of it's frequency is the actual time dilation. This effect is called as pure relativistic, as it is absent in the classical theory. It is quite clear and well-known fact in the special relativity. Note, that the distance between source and the 1st observer does not change in time, while being measured by the 1st observer.
All the issues, raised by the author, are due to the fact, that the author decided not to use α', but α as an angle, that is equal to π/2, in order to define transverse Doppler effect. It is obvious, that α is the angle between the wave direction and the relative motion of the 1st observer and source, but measured by another observer (let's call him 2nd observer). The 2nd observer in the rest with the light source. Then author substitutes α=π/2 to the formulas (1)-(2) and obtain a "blue" shift. The mistake of the author is that the formulas (1)-(2) are valid only if both the vector of the source motion (u, and β), and the angle between this vector and and wave direction (α'), are measured by the same observer. In case of author considerations, they are measured by different observers, situated at different reference frames.
If one wants to use 2nd observer to find the Doppler effect, for 2nd observer, his β=β₂=0, and his α'=α, formulas (1)-(2) will give ν'=ν₂'=ν. So there is no frequency shift for the 2nd observer. (as it should be for him, being in rest with the source).
For the 1st observer, substituting the β and α'=π/2 will give correct result: ν'=ν/γ, where γ=1/(sqrt(1-v²/c²)).
One should not mix this two cases, as the author does.
Let's look to the author's arguments, why one should use α=π/2 rather than α'=π/2 to define transverse...