Constructs lattice point sets with many rectangles and few isosceles triangles to produce explicit counterexamples to the Mizohata-Takeuchi conjecture for the paraboloid via transference principles.
Split primes and the Elekes-R\'onyai problem
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
abstract
There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left\{x+y+(x-y)^2:\ x, y \in A\right\}\right| \le|A|^{2-c}.$$ Since $x+y+(x-y)^2 \in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-R\'onyai problem. The proof combines two amplifications of the same local congruence defect: horizontal amplification over squarefree products of rational primes, and vertical amplification through bounded root-discriminant towers in which those primes split completely. In this way a fixed local density defect becomes macroscopic, producing a power saving. This phenomenon also suggests a broader mechanism for producing similar extremal constructions throughout combinatorics and number theory.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
Adapts known construction to prove existence of c>0 and large finite A subset R with |AA+A+A| << |A|^{2-c}, plus corollaries for other sum-product expressions.
citing papers explorer
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Rectangles, triangles and Schr\"{o}dinger waves
Constructs lattice point sets with many rectangles and few isosceles triangles to produce explicit counterexamples to the Mizohata-Takeuchi conjecture for the paraboloid via transference principles.
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More sum-product type counterexamples: products with shifts and $AA+A$
Adapts known construction to prove existence of c>0 and large finite A subset R with |AA+A+A| << |A|^{2-c}, plus corollaries for other sum-product expressions.