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Abstract

We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime \prime \prime }+a_2^{\prime \prime \prime })=(a_1^\prime +a_2^\prime )(a_1^{\prime \prime }+a_2^{\prime \prime })$. As applications, we improve results on difference-product/division estimates and on Balog–Wooley decomposition: For any finite subset $A$ of $\mathbb R$, $$\max \{|A-A|,|AA|\} \gtrsim |A|^{1+105/347},\quad \ \max \{|A-A|,|A/A|\} \gtrsim |A|^{1+15/49}.$$ Moreover, there are sets $B,C$ with $A=B\sqcup C$ such that $$\max \{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}.$$