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Creators/Authors contains: "Pohoata, Cosmin"

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  1. ; ; (, Inventiones mathematicae)
    Abstract Let$$p_{1},\ldots ,p_{n}$$ p 1 , … , p n be a set of points in the unit square and let$$T_{1},\ldots ,T_{n}$$ T 1 , … , T n be a set of$$\delta $$ δ -tubes such that$$T_{j}$$ T j passes through$$p_{j}$$ p j . We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points$$p_{1},\ldots , p_{n} \in [0,1]^{2}$$ p 1 , … , p n ∈ [ 0 , 1 ] 2 along with a line$$\ell _{j}$$ ℓ j through each point$$p_{j}$$ p j , there exist$$j\neq k$$ j ≠ k for which$$d(p_{j}, \ell _{k}) \lesssim n^{-2/3+o(1)}$$ d ( p j , ℓ k ) ≲ n − 2 / 3 + o ( 1 ) . It follows from the latter result that any set of$$n$$ n points in the unit square contains three points forming a triangle of area at most$$n^{-7/6+o(1)}$$ n − 7 / 6 + o ( 1 ) . This new upper bound for Heilbronn’s triangle problem attains the high-low limit established in our previous work arXiv:2305.18253. 
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    Free, publicly-accessible full text available March 14, 2026
  2. ; (, Duke Mathematical Journal)
    Free, publicly-accessible full text available February 15, 2026
  3. ; (, The Annals of Probability)
    Full Text Available 
  4. ; ; (, Discrete analysis)
    By using random multilinear maps, we provide new lower bounds for the Erd\H{o}s box problem, the problem of estimating the extremal number of the complete $$d$$-partite $$d$$-uniform hypergraph with two vertices in each part, thereby improving on work of Gunderson, R\"{o}dl and Sidorenko. 
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    Accepted Manuscript Full Text Available
  5. ; ; (, SIAM Journal on Discrete Mathematics)
    Full Text Available 
  6. ; (, Combinatorica)
    Full Text Available 
  7. ; (, Random Structures & Algorithms)
    Abstract Letfs, k(n)be the maximum possible number ofs‐term arithmetic progressions in a set ofnintegers which contains nok‐term arithmetic progression. For all fixed integersk > s ≥ 3, we prove thatfs, k(n) = n2 − o(1), which answers an old question of Erdős. In fact, we prove upper and lower bounds forfs, k(n)which show that its growth is closely related to the bounds in Szemerédi's theorem. 
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