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For the hair algae to grow that fast, I would think you have some decent nitrate and phosphate levels in the tank. A couple reactors (nitrate & phosphate media) would help but you might want to go a little more extreme and get those reduced a little more quickly.
Both of the above products should only be used with aggressive skimming (go wet skimming) and make sure your alkalinity is at about 10 or 11 before dosing as it will get a little lower. This causes no problem if you are aware of it and correct as needed.
While the combination of these two products with shipping are about $50 they can do the job of over $500 worth of media you would use in a reactor and will work a lot faster also. When you look at it that way it's pretty cheap.
The RO/DI unit will definitely be of help. The problem you might face is that the aquarium has a lot of phosphates stored up. You could be lucky however and they could be close to zero already because of all the hair algae growth.
If you don't already have a phosphate test kit pick one up. Do a test and see what it reads and then do another test about 4 days later to see if it rises (because the hair algae isn't using it anymore).
Since you mention your levels are zero I think your nitrates might be the limiting factor and you have an excess of phosphates (hopefully not the case).
Phosphates are pretty hard to get rid of. The best export is with macro algae or one of the products mentioned above. Phosphate sponge type media work well for keeping levels undetectable but once they build up this media can cost a small fortune to bring levels down.
Do the tests and if you find you have phosphates and don't mind a small trip you can take a ride to my house and I'll give you enough product to rid your system of the phosphates and/or nitrates. I could also bring some to the meeting if you plan on attending.
You'll probably have a small battle with the hair algae for a while but stay the course and you'll win!
Believe it or not, I have a phosphate test kit, and it's always read either 0 or the next step up on the test kit (whichever that is), even when testing the water before i put it in the tank.
I've heard silicates can cause the same type of problems, and that I have no test kit for, so it's really hard for me to guess at that one.
But, we'll see. I do have a fair amount of chaeto growing in my fuge...
As far as I understand, silicates are used only by diatoms - microalgae that are building silicate skeletons. vast majority are unicellular, though sometimes they could make small colonies and chains. They would cause algae blooms - green water. This, what you have, is almost certainly not caused by silicates.
For sure, you would most certainly see a diatom outbreak if you have an abundant amount of silicates. I'd say don't even bother to test for it if you don't have diatom problems.
If you do Red Sea makes a pretty good Silica test kit which is reasonably priced.
Carlo
PS If you register anything over zero on consumer phosphate test kits you have too much. If you register this on make up water then that's a problem for sure.
BTW, do you know what your TDS is of your make up water before adding salt?