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The Elekes-Szab\'{o} Problem and the Uniformity Conjecture

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Pith's one-line read Assuming the Uniformity Conjecture, this paper proves an improved Elekes–Szabó bound over the rationals: balanced intersections are $O(n^{9/5})$ in the worst applicable case and optimal $O(n)$ when the $y$-degree of $p$ is at least five.

desk verdict The theorem is proved correctly, but the abstract's balanced n^{9/5} claim is false for a legal example; the core mechanism is sound and the applications do control the missing M term. read the letter →

arxiv 2009.13258 v2 pith:53JP7P75 submitted 2020-09-28 math.CO math.AGmath.NT

Mehdi Makhul , Oliver Roche-Newton , Sophie Stevens , Audie Warren This is my paper · ORCID
classification math.COmath.AGmath.NT MSC 11G3052C1011B3014G05
keywords Elekes–SzabóproblemUniformityConjecturehyperellipticcurvesrationalpointsincidenceboundsadditivecombinatoricspinneddistancessumsofsquares
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a conditional improvement to the Elekes–Szabó problem over the rationals. For polynomials of the special form $F(x,y,z)=q(x,y,z)^2-p(x,y)$, it shows, assuming the Uniformity Conjecture, that the number of points of $Z(F)$ inside a Cartesian product $A\times B\times C\subset \mathbb Q^3$ is at most $O(n^{2-1/s})$, where $s\le 5$ depends only on the degree of $p$ in $y$. This improves the previous real-variable exponent $11/6$ to $9/5$ in the worst case, and gives the optimal linear bound when $\deg_y(p)\ge 5$. A reader should care because the same mechanism yields new conditional bounds for classical problems: triple intersections of three families of concentric circles, pinned distances from rational points, sums of squares, and arithmetic progressions in square sets.

What carries the argument

The load-bearing object is the hyperelliptic curve $Y^2=\prod_{i=1}^s p(a_i,X)$ attached to an $s$-tuple of distinct elements $a_1,\dots,a_s\in A$ whose polynomials $p(a_i,Y)$ have no common root. Because each $p(a_i,Y)$ is squarefree and $s d_p\ge 5$, this curve is smooth of genus $g=\lfloor(sd_p-1)/2\rfloor\ge2$; the Uniformity Conjecture then bounds by $B_g$ the number of its rational points. The proof arranges edges of a bipartite graph on $A\times B$ by common neighbourhoods, applies Hölder's inequality to reduce to counting $s$-tuples with large common neighbourhood, and uses this curve to show that in the generic $S_1$ part each $s$-tuple has at most $B_g$ common neighbours. The exceptional tuples are absorbed by the explicit error terms $|M_{A,p}|^{1/s}|B|$ and $L_F|C|$.

What would settle it

Search for a counterexample to the theorem's counting conclusion: take $F(x,y,z)=z^2-(x+y)$ (so $d_p=1$, $s=5$) and finite sets $A,B,C\subset\mathbb Q$ of size $n$ chosen so that $a+b=c^2$ has many solutions—for instance, sets of integers or rational squares constructed multiplicatively. If such sets satisfy $|Z(F)\cap(A\times B\times C)|\gg n^{9/5+\varepsilon}$ for any $\varepsilon>0$, then Theorem 3's bound is false; since the proof is conditional, that would either expose a flaw in the argument or disprove the Uniformity Conjecture.

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Extended reading notes

Core claim

The central claim is Theorem 3: under the Uniformity Conjecture, for any polynomial $F=q^2-p$ with $q\in\mathbb Q[x,y,z]$, $p\in\mathbb Q[x,y]$, $\partial F/\partial z\neq 0$, $\deg_y p=d_p\ge 1$, and $s$ the least integer with $s d_p\ge 5$, any finite $A,B,C\subset\mathbb Q$ with $p(a,y)$ squarefree for every $a\in A$ satisfy $|Z(F)\cap(A\times B\times C)| \ll_{s,d} |A||B|^{1-1/s}+|A|^{1-1/s}|B|+|M_{A,p}|^{1/s}|B|+L_F|C|$, where $M_{A,p}$ counts $s$-tuples of distinct $a$'s whose $p(a,y)$ share a complex root and $L_F$ counts pairs with $F(a,b,z)\equiv 0$. In balanced applications the first two terms dominate, yielding $n^{9/5}$ when $d_p=1$ and $O(n)$ when $d_p\ge 5$.

Load-bearing premise

The entire argument hinges on the Uniformity Conjecture: that for each genus $g\ge2$ there is a number $B_g$ such that every smooth genus-$g$ curve over $\mathbb Q$ has at most $B_g$ rational points; without such a uniform bound the key counting step only gives a bound that grows with the particular $a_i$ and the proof collapses.

Editorial extensions

If this is right

  • In the balanced case $|A|=|B|=|C|=n$, the worst applicable case $d_p=1$ gives $|Z(F)\cap(A\times B\times C)|\ll n^{9/5}$, improving the unconditional $n^{11/6}$ bound from the real-variable theorem.
  • When $\deg_y(p)\ge 5$, the theorem gives the optimal $O(n)$ bound, so the incidence count becomes linear.
  • Three families of concentric circles with rational centres and rational squared radii have $O(n^{5/3})$ triple intersection points in the balanced case, improving the previous $O(n^{11/6})$.
  • For any finite $P\subset\mathbb Q^2$ and two fixed rational points $p_1,p_2$, at least one pinned distance set has size $\gg |P|^{3/5}$.
  • A finite set $A$ of rational squares contains $O(|A|^{5/3})$ three-term arithmetic progressions, and sums of the form $A^{2k}+B^{2k}$ expand with rate $|B||A|^{1/s}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorem suggests that over $\mathbb Q$ the Elekes–Szabó degeneracy classification is strictly finer than over $\mathbb R$: polynomials such as $x+y+z^2$ are exceptional in the real theorem yet satisfy the new bound, so expansion is a rationality-sensitive phenomenon.
  • If the Uniformity Conjecture were proved with an effective $B_g$, every bound in the paper would become explicit; conversely, any combinatorial construction beating the $n^{2-1/s}$ bound would produce a family of hyperelliptic curves whose rational-point counts grow with the data, giving a counterexample to uniformity.
  • The same curve construction should generalize to $F=q^k-p$ for $k\ge3$, with genus at least two as soon as $s d_p\ge 2k+1$, yielding analogous $n^{2-1/s}$ bounds; the paper notes this extension but does not prove it.
  • The repeated-root hypothesis in Theorem 3 is doing real work: it is what guarantees the hyperelliptic curve is smooth, and the abstract statement omits it, so any application must check squarefreeness of $p(a,y)$ explicitly.
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Formalized claims in Lean

  1. Claim #1: The central claim is Theorem 3: under the Uniformity Conjecture, for any polynomial $F=q^2-p$ with $q\in\mathbb Q[x,y,z]$, $p\in\mathbb Q[x,y]$, $\partial F/\partial z\neq 0$, $\deg_y p=d_p\ge 1$, and $s$ the least integer with $s d_p\ge 5$, any finite $A,B,C\subset\mathbb Q$ with $p(a,y)$ squarefree for every $a\in A$ satisfy $|Z(F)\cap(A\times B\times C)| \ll_{s,d} |A||B|^{1-1/s}+|A|^{1-1/s}|B|+

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 2 minor

Summary. The paper proves a conditional Elekes-Szabó-type incidence bound, assuming the Uniformity Conjecture. Theorem 3 states that for F(x,y,z)=q(x,y,z)^2-p(x,y) with deg_y(p)=dp, if A,B,C are finite subsets of Q and p(a,Y) has no repeated roots for every a in A, then |Z(F) ∩ A×B×C| is at most |A||B|^{1-1/s}+|A|^{1-1/s}|B|+|M_{A,p}|^{1/s}|B|+L_F|C|, where s is the least integer with s dp ≥5, M_{A,p} counts s-tuples of distinct elements whose p(a_i,Y) have a common root, and L_F counts pairs with F(a,b,z) identically zero. The proof uses Hölder's inequality and reduces the main term to rational points on hyperelliptic curves of genus ≥2, where the Uniformity Conjecture applies. The paper then derives applications to triple intersections of concentric circles, pinned distances, sums of squares, expanders, additive energy, and graph-restricted sum-product problems, claiming in particular a balanced bound N^{9/5} that improves Raz-Sharir-de Zeeuw's N^{11/6}.

Significance. The conditional nature is explicit and the core reduction is genuine: the passage from incidences to rational points on hyperelliptic curves is non-trivial, clean, and gives interesting new conditional results. The Uniformity Conjecture input is clearly stated, and the paper is careful to define the exceptional terms M_{A,p} and L_F. In the geometric and arithmetic applications, the M term is controlled by case-specific calculations. However, the headline balanced bound stated in the abstract is not a consequence of Theorem 3 and is in fact false for a polynomial in the stated family, so the advertised improvement over Raz-Sharir-de Zeeuw needs substantial correction.

major comments (3)
  1. [Abstract; §2.1, Theorem 3 and following paragraph] The abstract claims that for F in the stated family and any finite sets A,B,C⊂Q with |A|=|B|=|C|=n, one has |Z(F)∩(A×B×C)| ≪ n^{2-1/s}. This is false for a legal choice of F. Take F(x,y,z)=z^2-y, so q=z, p=y, dp=1, s=5, and ∂F/∂z≠0. For every a, p(a,Y)=Y has the simple root 0, so the no-repeated-roots hypothesis holds. Let A=C={1,...,n} and B={1^2,...,n^2}; then (a,c^2,c)∈Z(F) for every (a,c)∈A×C, giving |Z(F)∩(A×B×C)|=n^2, which exceeds n^{9/5}. Theorem 3 itself is consistent because |M_{A,p}| ≈ |A|^5, so the term |M_{A,p}|^{1/s}|B| in (5) is ≈ n^2. The sentence after Theorem 3 stating that the first two terms of (5) are the important ones and that a balanced application gives N^{9/5} is therefore not justified by the theorem as stated. The abstract also omits the essential hypothesis that p(a,Y) has no repeated roots for all a∈A. The authors should correct the abstract and the paragraph after Theorem 3, and either keep the M term as an irreducible part of the theorem or add a hypothesis that guarantees M is small, such as requiring that p(a,Y) and p(a',Y) have no common root for distinct a,a' (as is done in the applications).
  2. [§2.3 and §5.1, Corollary 5] The statement of Corollary 5 in Section 2.3 says 'for any finite sets A,B⊂Q with |A|≤|B|', while the restated version in Section 5.1 and the proof require the stronger assumption 2d+1≤|A|≤|B|. The proof uses the condition |A|≥2d+1 to ensure that A\R is nonempty when the third term in (25) dominates; without it, the argument does not go through. As stated, the Section 2.3 version is stronger than what is proved and should be corrected to include the additional hypothesis, or the proof should be modified.
  3. [§3, S1 step (Eq. (14))] The proof of the S1 bound is sound, but it relies on the Uniformity Conjecture in a way that should be emphasized in the statement of the main theorem: the no-repeated-roots hypothesis on p(a,Y) is what guarantees that the auxiliary curve Y^2 = ∏ p(a_i,X) in Eq. (14) is smooth, hence of genus ≥2. Without this hypothesis the reduction to rational points fails. This is not an error in the proof, but it reinforces the point that the abstract's broad formulation, which drops this condition, is misleading.
minor comments (2)
  1. [§2.1] There is a typo in the sentence 'For some polynomials we attain the optimal upper bound |Z(F) × A × B × C| = ...': the expression should be the intersection |Z(F) ∩ (A × B × C)|.
  2. [Abstract] The abstract says that the integer s is dependent on the polynomial F, but in Theorem 3 s depends only on dp=deg_y(p), not on q or the rest of F; this is a minor clarity issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a genuine conditional reduction to the Uniformity Conjecture, with the auxiliary terms bounded by explicit arguments; the abstract's overbroad balanced claim is a correctness issue rather than a circular one.

full rationale

The central derivation is conditional on the Uniformity Conjecture, an external unproved input cited to Caporaso, Harris, and Mazur, and it is not used as a renamed version of the paper's own conclusion. In the proof of Theorem 3, the S1 step genuinely reduces the counting problem to rational points on the hyperelliptic curve Y^2 = prod_{i=1}^s p(a_i, X): each common neighbor b gives the rational point (b, prod_i q(a_i, b, c_i)), and the Uniformity Conjecture supplies a uniform bound on such points. The S2 term is bounded exactly by the definition of M_{A,p}, and S3 is bounded by the trivial degree bound and the number of non-distinct tuples; no parameter is fitted to the quantity being bounded. In every application, the extra term |M_{A,p}|^{1/s}|B| is controlled by explicit root-sharing arguments rather than by assumption: Corollaries 1 and 9 show each fixed element has at most four partners sharing a root, while Corollaries 5, 7, and 11 prove M_{A,p}=0 for the relevant polynomials. The self-citations [21] and [25] are not load-bearing for the main theorem. The abstract's advertised balanced n^{2-1/s} bound is indeed overbroad because Theorem 3 itself contains the term |M_{A,p}|^{1/s}|B|, and F(x,y,z)=z^2-y shows that this term can dominate; however, that is a defect in the stated corollary or abstract, not a case of the derivation reducing to its inputs. Thus no significant circularity is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests principally on the Uniformity Conjecture, an unproved external input, and on the explicit no-repeated-roots hypothesis; no number is fitted to data and no new entity is introduced. All constants in the upper bounds are either from the conjecture, from degree bounds on the polynomials, or from the size of the finite sets. The term B_g is an existential constant from the conjecture, not a free parameter.

assumptions (3)
  • domain assumption Uniformity Conjecture over Q: for every genus g ≥ 2 there exists an integer B_g such that every smooth curve of genus g over Q has at most B_g rational points.
    Load-bearing unproved input. Invoked in Section 3 to bound the S1 term via rational points on Y^2 = ∏ p(a_i, X), and in Corollary 8 to bound r_{A-B}(m) via Y^2 = X^k + m. Cited to Caporaso, Harris and Mazur [5]; no proof or effective value of B_g is given.
  • domain assumption For every a ∈ A, the univariate polynomial p(a, y) has no repeated roots; and for the S1 tuples, no two of the p(a_i, y) share a root in C.
    Hypothesis of Theorem 3 and of the S1/S2 decomposition in Section 3. It ensures ∏_{i=1}^s p(a_i, X) is square-free so the auxiliary curve is a smooth hyperelliptic curve of genus g = ⌊(s·dp - 1)/2⌋ ≥ 2. The abstract omits this requirement, though in applications it only excludes O_d(1) values of a (roots of the y-discriminant of p).
  • standard math Hölder's inequality, the genus formula for hyperelliptic curves, and standard degree bounds (e.g., L_F ≤ d^2 for irreducible F from [24, Lemma 2.1]).
    Used throughout Section 3 in equations (12), (14) and (16), and in the applications; all are standard results, not new postulates.

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Pith. "Pith review of The Elekes-Szab\'{o} Problem and the Uniformity Conjecture." pith.science (2026). https://pith.science/paper/53JP7P75

@misc{pith2026200913258,
  author       = {Pith},
  title        = {Pith review of: The Elekes-Szab\'o Problem and the Uniformity Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53JP7P75}},
  note         = {Machine review of arXiv:2009.13258}
}
abstract

In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of polynomials, and any finite sets $A, B, C \subset \mathbb Q$ with $|A|=|B|=|C|=n$, we have \[ |Z(F) \cap (A\times B \times C)| \ll n^{2-\frac{1}{s}}. \] The value of the integer $s$ is dependent on the polynomial $F$, but is always bounded by $s \leq 5$, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw (arXiv:1504.05012). We give several applications to problems in discrete geometry and arithmetic combinatorics. For instance, for any set $P \subset \mathbb Q^2$ and any two points $p_1,p_2 \in \mathbb Q^2$, we prove that at least one of the $p_i$ satisfies the bound \[ | \{ \| p_i - p \| : p \in P \}| \gg |P|^{3/5}, \] where $\| \cdot \|$ denotes Euclidean distance. This gives a conditional improvement to a result of Sharir and Solymosi (arXiv:1308.0814).

Figures

Figures reproduced from arXiv: 2009.13258 by the authors.

Figure 1
Figure 1. Three families of concentric circles. A triple intersection po [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    N. Alon, O. Angel, I. Benjamini, and E. Lubetzky, Sums and products along sparse graphs , Israel J. Math. 188, 353–384 (2012)

    work page 2012

  2. [2]

    N. Alon, I. Z. Ruzsa and J. Solymosi, Sums, products, and ratios along the edges of graph , Publ. Mat. 64, no. 1, 143-155 (2020)

    work page 2020

  3. [3]

    Balog and E

    A. Balog and E. Szemer´ edi, A statistical theorem of set addition , Combinatorica 14, 263-268 (1994)

    work page 1994

  4. [4]

    Bombieri and U

    E. Bombieri and U. Zannier, A note on squares in arithmetic progressions. II. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl.13, No.2 , 69-75 (2002)

    work page 2002

  5. [5]

    Caporaso, J

    L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points , J. Amer. Math. Soc. 10, no. 1, 1–35 (1997) 21

    work page 1997

  6. [6]

    Chang, On problems of Erd˝ os and Rudin, J

    M.-C. Chang, On problems of Erd˝ os and Rudin, J. Funct. Anal. 207, 444–460 (2004)

    work page 2004

  7. [7]

    Cilleruelo and A

    J. Cilleruelo and A. Granville, Lattice points on circles, squares in arithmetic progressi ons and sumsets of squares, CRM Proceedings and Lecture Notes: Additive Combinatorics, AMS a nd Centre de Recherches Math´ ematiques, pp. 241–262 (2007)

    work page 2007

  8. [8]

    Darmon and L

    H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s Last Theore m, Journal f¨ ur die reine und angewandte Mathematik, no. 490, 81-1 00 (1997)

    work page 1997

Show all 32 references
  1. [9]

    Elekes, Circle grids and bipartite graphs of distances , Combinatorica 15, 167-174 (1995)

    G. Elekes, Circle grids and bipartite graphs of distances , Combinatorica 15, 167-174 (1995)

    1995

  2. [10]

    Elekes, SUMS versus PRODUCTS in Number Theory, Algebra and Erd˝ os Ge ometry, Paul Erd˝ os and his Mathematics II, Bolyai Society Mathematical Studie s 11, 241–290 (2002)

    G. Elekes, SUMS versus PRODUCTS in Number Theory, Algebra and Erd˝ os Ge ometry, Paul Erd˝ os and his Mathematics II, Bolyai Society Mathematical Studie s 11, 241–290 (2002)

    2002

  3. [11]

    Elekes and L

    G. Elekes and L. R´ onyai, A combinatorial problem on polynomials and rational functi ons, J. Combin. Theory Ser. A. 89, 1-20 (2000)

    2000

  4. [12]

    Elekes and E

    G. Elekes and E. Szab´ o, How to find groups? (And how to use them in Erd˝ os Geometry?) , Combinatorica 32 (5) (2012) 537–571

    2012

  5. [13]

    Erd˝ os, L

    P. Erd˝ os, L. Lov´ asz and K. Vesztergombi, On the graph of large distances , Discrete Comput. Geom. 4, no. 6, 541-549 (1989)

    1989

  6. [14]

    Erd˝ os,Ulam, the man and the mathematician , J

    P. Erd˝ os,Ulam, the man and the mathematician , J. Graph Theory, 9 (4) (1985) 445–449. Also appears in in Creation Math. 19, 13-16 (1986)

    1985

  7. [15]

    Faltings, Erratum: Finiteness theorems for abelian varieties over nu mber fields , Invent

    G. Faltings, Erratum: Finiteness theorems for abelian varieties over nu mber fields , Invent. Math. 75 (1984), no. 2, 381

    1984

  8. [16]

    Fox and C

    J. Fox and C. Pohoata, Sets without k-term progressions can have many shorter prog ressions, arXiv preprint arXiv:1908.09905 (2019)

    2019 arXiv

  9. [17]

    Galbraith, Mathematics of Public Key Cryptography

    S. Galbraith, Mathematics of Public Key Cryptography . Cambridge University Press (2012)

    2012

  10. [18]

    M. Z. Garaev and C.-Y. Shen, On the size of the set A(A + 1), Math. Z. 265, no. 1, (2010) 125-132

    2010

  11. [19]

    Hanson, O

    B. Hanson, O. Roche-Newton and D. Zhelezov, On iterated product sets with shifts II , To appear in Algebra Number Theory

  12. [20]

    van Luijk, On perfect cuboids , Doctoraalscriptie, Mathematisch Instituut, Universiteit Utrecht, Utrecht (2000)

    R. van Luijk, On perfect cuboids , Doctoraalscriptie, Mathematisch Instituut, Universiteit Utrecht, Utrecht (2000)

    2000

  13. [21]

    Makhul, O

    M. Makhul, O. Roche-Newton, A. Warren and F. de Zeeuw, Constructions for the Elekes-Szab´ o and Elekes-R´ onyai problems, Electron. J. Combin. 27, no. 1, Paper 1.57 (2020)

    2020

  14. [22]

    W. G. Nowak and W. Recknagel, The distribution of pythagorean triples and a three dimensi onal divisor problem , Math. J. Okayama Univ. 31, 213-220 (1989)

    1989

  15. [23]

    O. E. Raz, M. Sharir, J. Solymosi, Polynomials vanishing on grids: The Elekes-R´ onyai problem revisited, Amer. J. Math. 138(4):1029-1065 (2016) 22

    2016

  16. [24]

    O. E. Raz, M. Sharir and F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes–Szab´ o theorem revisited, Duke Math. J. 165 (2016), no. 18, 3517–3566

    2016

  17. [25]

    Roche-Newton and A

    O. Roche-Newton and A. Warren, Improved bounds for pencils of lines , To appear in Proc. Amer. Math. Soc

  18. [26]

    Rudin, Trigonometric series with gaps, Journal of Mathematics and Mechanics

    W. Rudin, Trigonometric series with gaps, Journal of Mathematics and Mechanics. 9, no. 2, 203–227 (1960)

    1960

  19. [27]

    Shaffaf, A solution of the Erd˝ os-Ulam problem on rational distance s ets assuming the Bombieri–Lang conjecture, Discrete Comput

    J. Shaffaf, A solution of the Erd˝ os-Ulam problem on rational distance s ets assuming the Bombieri–Lang conjecture, Discrete Comput. Geom., 60 (8) (2018)

    2018

  20. [28]

    Sharir and J

    M. Sharir and J. Solymosi, Distinct distances from three points , Combin. Probab. Comput. 25, no. 4, 623-632 (2016)

    2016

  21. [29]

    I. D. Shkredov, Some remarks on sets with small quotient set , Mat. Sb. 208, no. 12, 144-158 (2018)

    2018

  22. [30]

    I. D. Shkredov and J. Solymosi, The Uniformity Conjecture in additive combinatorics , arXiv preprint arXiv:2005.11559 (2020)

    2020 arXiv

  23. [31]

    Solymosi, Expanding polynomials over the rationals , arXiv preprint arXiv:1212:3365v1 (2012)

    J. Solymosi, Expanding polynomials over the rationals , arXiv preprint arXiv:1212:3365v1 (2012)

    2012

  24. [32]

    T. Tao, The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture , blog post, 20 December 2014, https://terrytao.wordpress.com/2014/12/20/the-erdos -ulam-problem-varieties-of-general-type-and-the-bom bieri-lang-conjecture/ M. Makhul, F akult¨at f ¨ur...

    2014

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