Description of problem:
The other day, I was in #Fedora-SELinux solving this.

A very nice guy there came out with the following solution:

# cat mythin.te; echo; echo; cat mythin.fc 
policy_module(mythin, 1.0.0)
gen_require(` type thin_t, thin_var_run_t; ')
manage_sock_files_pattern(thin_t, thin_var_run_t, thin_var_run_t)
apache_read_sys_content(thin_t)
gen_require(` type httpd_t; ')
stream_connect_pattern(httpd_t, thin_var_run_t, thin_var_run_t, thin_t)


/var/run/thin(/.*)? gen_context(system_u:object_r:thin_var_run_t, s0)

I can confirm that this works; but I dunno if it's the best approach.

Usually, you will have your sinatra apps self contained; with certain configs there and all. 
SELinux is preventing /usr/sbin/nginx from 'write' accesses on the sock_file btcsrv.sock.

*****  Plugin catchall (100. confidence) suggests  ***************************

If you believe that nginx should be allowed write access on the btcsrv.sock sock_file by default.
Then you should report this as a bug.
You can generate a local policy module to allow this access.
Do
allow this access for now by executing:
# grep nginx /var/log/audit/audit.log | audit2allow -M mypol
# semodule -i mypol.pp

Additional Information:
Source Context                system_u:system_r:httpd_t:s0
Target Context                unconfined_u:object_r:var_run_t:s0
Target Objects                btcsrv.sock [ sock_file ]
Source                        nginx
Source Path                   /usr/sbin/nginx
Port                          <Unknown>
Host                          (removed)
Source RPM Packages           nginx-1.2.9-1.fc18.x86_64
Target RPM Packages           
Policy RPM                    selinux-policy-3.11.1-96.fc18.noarch
Selinux Enabled               True
Policy Type                   targeted
Enforcing Mode                Enforcing
Host Name                     (removed)
Platform                      Linux (removed) 3.9.3-201.fc18.x86_64 #1 SMP Tue
                              May 21 17:02:24 UTC 2013 x86_64 x86_64
Alert Count                   2
First Seen                    2013-05-28 04:32:04 CDT
Last Seen                     2013-05-28 04:32:04 CDT
Local ID                      48a17b07-ebfb-43ca-a011-86b1fdc74ee1

Raw Audit Messages
type=AVC msg=audit(1369733524.220:499): avc:  denied  { write } for  pid=21214 comm="nginx" name="btcsrv.sock" dev="tmpfs" ino=138342 scontext=system_u:system_r:httpd_t:s0 tcontext=unconfined_u:object_r:var_run_t:s0 tclass=sock_file


type=SYSCALL msg=audit(1369733524.220:499): arch=x86_64 syscall=connect success=no exit=EACCES a0=b a1=7f04ff67c7e0 a2=6e a3=7fff1e5ec5f2 items=0 ppid=21213 pid=21214 auid=4294967295 uid=990 gid=986 euid=990 suid=990 fsuid=990 egid=986 sgid=986 fsgid=986 ses=4294967295 tty=(none) comm=nginx exe=/usr/sbin/nginx subj=system_u:system_r:httpd_t:s0 key=(null)

Hash: nginx,httpd_t,var_run_t,sock_file,write

audit2allow

#============= httpd_t ==============
allow httpd_t var_run_t:sock_file write;

audit2allow -R
require {
	type var_run_t;
	type httpd_t;
	class sock_file write;
}

#============= httpd_t ==============
allow httpd_t var_run_t:sock_file write;


Additional info:
reporter:       libreport-2.1.4
hashmarkername: setroubleshoot
kernel:         3.9.3-201.fc18.x86_64
type:           libreport