The last comment is the one to note.....High performance summer tyres are rubbish at temperatures below 7 C for stopping distances, all around grip in wet and dry and in icy/snowy conditions are a total liability. Pirelli P Zero as shod on the S3 are not guaranteed below 7 C for cracking etc.
All season high performace tyres which are in short supply in the UK, but not in North America, will give you good performance in most winter conditions including light snow. The ones of note are the Pirelli P zero all season plus and the Continental DWS-06