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Just noticed this, too.
Doing a search on it on google (biw parameter google) it turned out to be the width of the browser in pixels.
Interesting how it varies. Small snippet from the logs for biw values:
1016, 851, 778, 1004, 772, 780, 1003, 780, 1024, 839, 995, 1007, 1005, 995, 993, 759, 740, 1004 (and dozens more)
Care to optimize your page for all of these widths?
You've seen the random spaces in URLs that are displayed? They are so that long URLs don't cause horizontal scrolling on a small screen.
I think that it is a key part of any such decision to monitor the utilisation of such features and whether demand continues to be justified.
Adjusting the value of biw manually seems to have a very slight effect on the page layout, mainly on the blue bar containing the top two sponsored results.
For example setting biw to 200 makes the blue bar go all the way across the page to the right hand side pushing the rest of the sponsored links down below it, but changing it to 1024 makes the blue bar stop before the right hand side column and the right hand side sponsored links start from the top.
This all depends on you entering a query where there are sponsored results at the top.