Applying for postdoc in a mathematics department where no one has similar research interests I'm a PhD student in mathematics, working in a quite unpopular field (recursion theory). As I'm now applying for postdoctoral positions in the US, I've encountered the following problem: The number of mathematics departments in the US having notable researchers in my field is quite small. As I'm quite flexible with my research interests, I added departments with researchers in related fields (such as model theory and set theory) to my list. But even then, there are less than 20 relevant departments across the US to which I can apply. 
I heard that in order to maximize the chances of getting a postdoctoral position, mathematicians often apply to 50 positions or even more. I'd like to do the same thing, but I have some questions about it:


Are there any arguments against applying for postdoc positions in math departments where your research interests are not even remotely represented?
What should I do in order to increase my chances of getting a postdoc position in such a department? More specifically, what should I write in my cover letter (where applicants usually mention relevant faculty members, and sometimes also explain why they fit the position)? Should I contact a faculty member? What should I write to him/her? Any other useful advice?
Assuming that I have a strong application, do I have a reasonable chance of getting a position in such departments?
Bottom line question: Should I invest time and effort applying for positions in such departments?
I sympathize with your situation - recursion theory as a whole is essentially the size of a sub-subfield in other areas of mathematics. Of the relatively small number of U.S. departments with logic programs, many are also extremely selective, and so getting a postdoc in any area of logic is challenging.  It certainly cannot hurt to apply to every program with a logic group that is hiring a postdoc, although if they are not looking for a recursion theorist this year your chances will be much lower. 

On the other hand, the recursion theory community has at least one advantage due to its small size: many of the researchers in the U.S. know each other, and know each others' students.  I have always thought it is a friendly community, at least compared to the rumors about some other fields. 

Two more things that are also worth note.


If you cannot get a true "postdoc", it is sometimes possible to find at least a temporary position at an institution with a logic program. Some of the logic departments are quite large, and may have another round of hiring for instructors later in the spring or summer. So, if you don't have a postdoc offer by the expected date, make sure you reach out (and/or ask your advisor for advice) to schools where you might be able to work while you apply again next year. It is particularly helpful in this case to try to get more research done in the spring, before you graduate, to strengthen your applications for the following year. I find it is always hard to get research done in the first fall at a new institution, while also applying for jobs. 
Depending on the type of department you are looking for, you may want to consider broadening your search to less-famous schools. If you are interested in both excellent teaching and excellent research, by doing some internet legwork you can often find schools that are not prestigious but happen to have strong research programs (relative to the kind of school). These schools are often less picky about the "area" you work in, and you might be able to sell yourself as a discrete mathematician or mathematical computer scientist. Ironically, you sometimes have more research flexibility at these schools, for various reasons, which is a good thing if you are a recursion theorist. So look for choice tenure-track openings as well as postdocs. Just be well prepared when you give an interview - tenure track interviews are not like postdoc interviews.