The truth is that you can find as much mathematics as you want in any of these fields. They all lend themselves to mathematical treatment, and consequently, if you are mathematically inclined, you will find the math.
There is a tendency to try to sway students into particular areas of study by suggesting that certain areas have more prestige than others (this is nonsense!). For those who enjoy math, telling them that there is more math in field X than there is in field Y is a way to try to turn them toward X and away from Y. This is an unfortunate manipulation of young people who do not know enough about any of these fields to make wise choices. This happened to me many years ago, and it cost me about 3 semesters of college before I got back to where I belonged.
The same manipulation is done with the word "research" and again, it is unfortunate.
The key thing is to recognize the difference between engineering and science. The word "science" comes from a Latin root that means "knowledge" and it is about the pursuit of new knowledge, things not previously known to mankind. In contrast, engineering is about creating new and useful systems and machines for mankind. Thus, the question to ask yourself is, "Do you want to pursue knowledge for its own sake, or do you want to develop useful things for people?"
Many people have the false impression that with a physics degree they can just slide over and practice engineering. In actual fact, it does not work that way. The difference in education is real, so don't plan to study one thing but do the other in your later work life.