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.
arXiv preprint arXiv:2007.03528 , year=
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erd\H{o}s on arithmetic progressions.
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2026 5representative citing papers
A refinement of Chang's lemma that adds cosetwise l1 control on correlations outside the large spectrum subspace, producing a localized counting lemma for subsets of finite abelian groups.
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.
A witness-split and window-pruning SAT framework finds no 44-element 3-AP-free subset of [1,212] but leaves two resistant instances unsolved.
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A strengthening of Chang's lemma
A refinement of Chang's lemma that adds cosetwise l1 control on correlations outside the large spectrum subspace, producing a localized counting lemma for subsets of finite abelian groups.
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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.