Speaking directly to your question about improving your position on search engines:
The answer is no, Twitter will not affect your position.
Here’s why: links on Twitter have the rel=“nofollow” attribute, which means search engines don’t follow links on Twitter, so you could post 500 tweets (with links to your site) every day and Google wouldn’t care.
However… it could potentially be used to indirectly and negligibly affect rankings for a company blog or site, by having an embedded feed from your Twitter account on your blog or home page. What happens is when you post a tweet on Twitter, it also shows up as a couple of sentences of new content on your blog or site, within the little widget on that page. And fresh content is one of several hundred things that search engines like to see as evidence of a useful site in general terms. But this would only affect sites that were considered extremely high quality already, and crawled extremely frequently in the first place, otherwise if your site is only crawled once a month, tweeting all day long won’t accomplish anything because the crawlers aren’t already coming through enough to notice!
Anyway, forget about using Twitter to help your search rankings. There are dozens of important things you need to be doing for SEO, and this isn’t one of them.
(Now… as noted in the other comments, you could use Twitter (depending on your industry) as a general marketing tool. If your market hangs out on Twitter, and if you’ve got time to worry about it, having an account might be useful. It could refer people to your real site. It can allow feedback from clients or prospects in a casual setting. It can allow the fast spreading of bits of news and info. It can help establish your business as a leader in knowledge, innovation, or customer service. So to help rankings – no; to help with general marketing – perhaps.)