I was thinking about your post today while I was driving and a couple of thoughts occurred to me:
First, do you know what cycle day you ovulated by chance? Because how many days past ovulation you are could really effect beta #s. If your beta is 4000+, then you're probably around 20 days past ovulation or so, so about 5 weeks gestation.
Second, any chance there could have been twins? I was just thinking about how close you were to doubling, even though it was 3 days and not 2 days. Sometimes if you had the beginning of a twin pregnancy, and then one of the babies dies; your betas can slow down. So if you had 2 babies there for little while, and say one died in between these two beta draws you had; then your number would not double. I don't know if that's making any sense.
I read somewhere that a lot more twins are conceived than people think; and one of the babies dies before it is ever discovered. Most women do not get betas drawn, and do not have ultrasounds before 8 weeks. (that's mainly reserved for those of us with infertility and recurrent pregnancy loss.)
..and you're right, after your beta reaches 2000 the pregnancy should be visible by ultrasound. However, at this point you'd probably only be able to see a fetal pole and yolk sac. It might be best to wait another week so that you can (hopefully!) see a heartbeat. That will give you a lot of peace of mind.
I'm not a doctor, I've just spent a very long time dealing with infertility/recurrent pregnancy loss and seeing a Reproductive Endocrinologist. I learned a lot. =)
I'd love to help in any other way I can!