How important is Social Media for building domain authority / Google rankings? Are there any cases?

[Seeders]

I really would like to know if someone tested the importance of Social Media for Google rankings.
Are there some sites who build authority only by doing good social media? 
Ofcourse, I know it is all about the mix (content, linkbuilding, social media, etc.) but how important is it?  
I know many sites who rank good without any form of social media, but I do not know any sites who do only social media and rank high.

I hope there are some good cases which give good insight.

ps. I know it becomes more and more important...

[ChristopherGlaeser]

To date my site does not get a lot of user engagement via G+, but I am helping those who write for our site with their G+ accounts.  Our writers are experts and practitioners in our niche and they are not (nor do I expect them to be) SEO experts.  I explain to them the potential benefits of G+ and help them set up rel=author to get credit for both the articles they write for our site and also the articles they write for other sites.  I also encourage them to increase their circle.  I'm confident our readers would prefer to read articles by experts in our field rather than ghost written articles.  I also compensate them better than ghost writers.  I think it's a win/win if these experts' photos are in the SERP and they get the recognition they deserve and it should improve the CTR as well.

[McTaggart]

I tend to work in Britain and Ireland and use Google+, Facebook, Twitter.

Then additional tools depending on site.

I see same probs with Google+ but even if engagement isn't so good (yet!) there are plenty of other benefits - some are mentioned here: http://socialmediatoday.com/node/1600736

Social signals are important: http://moz.com/blog/ranking-factors-2013

[Seeders]

Thanks Christopher and Luke. If I understand it correctly, it might help a bit,.. but it does not pay off big time
Luckely the social traffic is also important on it's own.
Which social media do you use?

I am pretty sure Google+ becomes more and more important. But here in the Netherlands nobody uses it, except SEO marketeers How is this in the USA @Christopher Glaeser? And Luke, in your country?

[ChristopherGlaeser]

I have a niche site with modest organic search traffic so perhaps this is the exception rather than the rule, but since I've made an effort to engage using social, the social traffic now outperforms the organic.  The organic is up a bit, no doubt because of the surge in social, but the social seems to be paying off as an end to itself rather than a way to increase organic search.

[McTaggart]

I've just been playing with one of my own new sites. Level of social media engagement appears to make the difference between a top of first page ranking and second page ranking re: a competitive keyphrase I'm tracking. I am going to widen my testing to additional keyphrases in due course.

Btw, you might find this interesting: http://moz.com/blog/ranking-factors-2013