Any set avoiding a genus-one translation-invariant linear equation over finite fields can be improved upon by a higher-dimensional avoiding set with strictly higher normalized size.
Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields
5 Pith papers cite this work. Polarity classification is still indexing.
5
Pith papers citing it
abstract
We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem).
years
2026 5representative citing papers
w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
Constructs a line-free set in F_p^3 of size (p-1)^3 + (1/8)p^{3/2} - O(p), the first superlinear improvement over the hypercube.
The paper establishes the existence of positive constants c and c_IP for the IP Szemeredi theorem over finite fields and gives strong quantitative bounds in the special cases of Roth and IP-Roth theorems.
The author proves f(N) ≫ N exp(−(2√(log(24/7))+o(1))√(log log N)) for the largest harmonic-triple-free subset of [N], matching the form of the best 3-AP-free lower bound with log N replaced by log log N.
citing papers explorer
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Beating Product Constructions for Linear Equations Over Finite Fields
Any set avoiding a genus-one translation-invariant linear equation over finite fields can be improved upon by a higher-dimensional avoiding set with strictly higher normalized size.
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Three-color van der Waerden numbers grow super-exponentially
w(k;3) > 2^{k (log^* k)/4} for large k, so the three-color van der Waerden number grows super-exponentially.
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A superlinear improvement on line-free sets in $\mathbb{F}_p^3$
Constructs a line-free set in F_p^3 of size (p-1)^3 + (1/8)p^{3/2} - O(p), the first superlinear improvement over the hypercube.
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On the Furstenberg-Katznelson constant for the IP Szemeredi theorem over finite fields
The paper establishes the existence of positive constants c and c_IP for the IP Szemeredi theorem over finite fields and gives strong quantitative bounds in the special cases of Roth and IP-Roth theorems.
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Large Sets of Integers with No Harmonic Triples
The author proves f(N) ≫ N exp(−(2√(log(24/7))+o(1))√(log log N)) for the largest harmonic-triple-free subset of [N], matching the form of the best 3-AP-free lower bound with log N replaced by log log N.