Fourier convolution of a histogram
Actually, it does not really matter, if a or b are histograms or time domain sequences. However, I suppose you want to convolve to histograms to get the probability distribution of a random variable that is the sum of two variables. Then, its PDF is the convolution of the PDF of the original variables. But, considering the convolution of two time-domain sequences might make it easier to understand. 
The convolution theorem states fft(ab)=fft(a)*fft(b), where * denotes the N-point circular convolution and a and b are of length N points. Your statement of the convolution theorem fft(ab) = fft(a) . fft(b) is not correct.
As I said, the convolution theorem in discrete domain corresponds to circular convolution. Most probably, you want to have a linear convolution. Then, you need to zero-pad a and b before applying the convolution theorem. 
If the original bins range from 0-3 (left-side of the interval), then the bin edges after convolution would range from 0-6, because the (linear) convolution has a length of (N+N-1) samples, where N is the length of the sequence (which is 4 in your case). You can also understand this from the fact that the convolution of the histogram is the PDF of the sum of both random variables: The minimum sum would be 0, the maximum sum would be 3+3=6.


For more information on circular convolution, you might have a look at an article I wrote or the Wikipedia entry.