4.a3 move in Four Knights Game

1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.a3!?
The so called Gunsberg Variation
4.a3 is definitely not a wasting move that amateurs play without blinking, for it has many purposes. As we can see it actually puts Black in some kind of "zugzwang".. 4...Bc5 fails to simple Nxe5 trick followed by d4 and intermezzo 5...Bxf2+ isn't that good. Black can try to prove that 4.a3 was a waste of time by playing 4...d5, however that is not the case, because after 5.Bb5 White is clearly fine. I actually have a game against masterhec (2400) where Black played 4...a6 and I followed with 5.d4 with good game. This is for players who wants to try something new..Also, you will definitely almost always be in advantage over your opponent theory-wise.

What about the quiet 4...d6 or 4...Be7 and black continues with development, the f8 bishop moving to e7 instead of b4?
That's probably what I would do if my opponent played 4.a3.

Well 4...Be7 doesn't present White any problems and White can continue with 5.d4 and it's questionable if Black's DSB is placed well, while 4...d6 locks the DSB inside. These moves are playable ofcourse, but nothing for White to worry about.

If we actually follow the main line of 4...d6, then after 5.d4 Bg4 6.d5 Nd4 7.Be3 Nxf3 8.gxf3 Bd7 White has a space advantage and the open g-file he can put to a good use.

Actually I am a big expert in this line and scored many important points.

Example: semi-final of the Cup in my federal state, Sarg0n white

I am glad to hear that Sarg0n.

The famous SOS series in 15 volumes with 20 chapters each. It is Vol. 1 / chapter 1. Now you can estimate the value. ;)

Yes, indeed hah. 4.a3 is actually a really nice opening surprise.

Well played btw! Black lost pretty convincingly.