They researched the prevalence of cancer in ancient societies using a variety of methods - remains, reporting, etc. The goal was to determine whether the relative lack of cancer in ancient societies was driven by some sort of external factor such as inability to detect cancer in remains, poor reporting, lower life expectancies, etc or rather it was simply because cancer was simply much less common. The evidence available tends to point very much towards the latter which, in turn, suggests that we have created environmental conditions which are driving cancer. The paper references two papers which have pointed to general pollution as a major potential driver there.
Death rates are not a relevant measurement on this topic either since they're heavily confounded by increase in survival rates. You need to look at incident rates, which continue to increase worldwide. Even lung cancer is now on the rise among females.
How does this paper account for the increase in life expectancy? Many types of cancer appear in people of older age and reaching that age was not nearly as common in ancient societies.
Two ways. By looking at the causes of death of those that did live to old age as well as by considering cancers (such as bone) that tend to occur disproportionately in young people.
Literally the only thing they seem to say for sure in that paper is that they expect to see bone evidence they don't see. While I don't doubt the earnestness of the paper's authors, their primary study seems to be in Egyptology first. They practically gush about how the ancient Egyptian medicines are still in use today...but then only list ones based on papyrus. They don't account for the fact that our sedentary lifestyle is shown to increase cancer rates, they barely explore different cultures, with only a passing reference to Greece, who has numerous treatments, including excision to remove cancer, and admit that the ancient Egyptians didn't even classify benign tumors differently than no-benign. They don't account for the increased consumption of meat in a western diet. They don't account for massively higher numbers of humans.
Based on their paper alone I can think of at least 5 other hypothesis that they didn't even cover.
1. Periods of fasting and lack of food decrease cancer.
2. Less sedentary lifestyles decrease cancer risks(this has actually been shown to have some promise, with HIIT workouts showing improved response rates and decreased reccurance)
3. Decreased infant mortality caused an increase in susceptibility to cancer because weaker physical specimens are surviving.
4. Firearms. Cancer rates have increased with the output of firearms.
5. Decreased prayer to the Egyptian gods. As these gods declined cancer rates increased.
Some of these are obviously less serious than others, however it doesn't excuse the plausible possibilities. These aren't oncologists, their paleontologists, also known as historians who guess about things based on what they dig out of the ground.
To sum up - cancer is driven by our environment, not just random mutations. As commented by others - we are doing a lot more to our environment and food sources to drive cancer (and other health problems) than random chance.
They researched the prevalence of cancer in ancient societies using a variety of methods - remains, reporting, etc. The goal was to determine whether the relative lack of cancer in ancient societies was driven by some sort of external factor such as inability to detect cancer in remains, poor reporting, lower life expectancies, etc or rather it was simply because cancer was simply much less common. The evidence available tends to point very much towards the latter which, in turn, suggests that we have created environmental conditions which are driving cancer. The paper references two papers which have pointed to general pollution as a major potential driver there.
Death rates are not a relevant measurement on this topic either since they're heavily confounded by increase in survival rates. You need to look at incident rates, which continue to increase worldwide. Even lung cancer is now on the rise among females.
[1] - https://www.nature.com/articles/nrc2914