Here is the reasoning I was given in an email from LSE explaining why I wasn’t given an offer.
“The feedback we have received from the Admissions Selector indicates that your application was unsuccessful as a consequence of the qualities of your personal statement. As competition for places at LSE is intense, the Personal Statement is considered carefully when deciding whether to make an offer. Your personal statement did not do enough to convince the Admissions Selector of your interest in all of the subjects offered on the combined programme to which you have applied when compared to the Personal Statements of similar applicants.”
As for my personal statement, it was more heavily skewed towards Econ than maths since Cambridge only offers pure economics. The way I tried to go about it was to talk about my strong foundation in maths (my passion for it, UKMT Olympiad results, etc), and mentioned multiple times “quantitative and statistical methods” used in economics. I also mentioned my IB extended essay, which I wrote for maths on game theory.
Frankly I don’t really see how I could have leaned into maths anymore without diverting from economics and hurting my chances for pure maths. I guess what LSE wanted to see was more of an interest in pure maths and maths not so closely related to economics or social sciences. E.g. all the books I mentioned were about economics, not maths. You could mention a maths focused book you read, some extra maths course you did, etc.
That being said, LSE was the only uni from my choices to not give me an offer. E.g. Warwick was happy to take me for MORSE (very maths heavy), and both UCL and Edinburg gave me offers for Economics and Statistics. So either my grades etc we’re just very strong, or the personal statement was balanced well enough for those unis to give me an offer.