A new combinatorial large sieve produces the first super-polylogarithmic upper bounds of the form N exp(-c log N / log log N) for Sidon sets in squares and no-repeated-distance sets in the grid.
Split primes and the Elekes-R\'onyai problem
3 Pith papers cite this work. Polarity classification is still indexing.
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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 3verdicts
UNVERDICTED 3representative citing papers
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.
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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A combinatorial large sieve for Sidon sets, distances, and norm forms
A new combinatorial large sieve produces the first super-polylogarithmic upper bounds of the form N exp(-c log N / log log N) for Sidon sets in squares and no-repeated-distance sets in the grid.
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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.