Furstenberg–Sárközy exponent for square-difference-free sets
Let r(N) be the maximum size of a subset A⊂\1,…,N\ with no non-zero square differences a-b=n^2.Then C_4b is the least constant such that r(N) ≤ N^C_4b+o(1).
0claims
0verified
Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
All fields Mathematics 936Computer science 28Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
1011 shown· page 4 of 21
Gagliardo-Nirenberg Inequality
Let 1 ≤ q ≤ ∞, and let j and m be non-negative integers such that j < m. Furthermore, let 1 ≤ r ≤ ∞, p ≥ 1 be real and θ ∈ [0, 1] such that the following relations hold: 1/p = j + θ ( 1/r - m ) + 1 - θ/q, j/m ≤ θ < 1.
0claims
0verified
Gauss circle problem exponent
Let N(t) := \#(m,n)∈ℤ^2: m^2+n^2≤ t^2 be the number of integer lattice points inside the (closed) disk of radius t centered at the origin. The Gauss circle problem is to find the smallest exponent θ such that, for every ε>0, N(t) = π t^2 + O(t^θ+ε).
0claims
0verified
Gilbert-Pollak conjecture (Steiner ratio)
C_43 is defined as the infimum of the ratio of the length of the Steiner Minimal Tree to the length of the Euclidean Minimum Spanning Tree over all finite sets of points V ⊆ ℝ^2: C_43 = inf_VL_S(V)/L_M(V), where L_S(V) and L_M(V) denote the lengths of Steiner Minimal Tree and Minimum Spanning Tree…
0claims
0verified
GL_2 Ramanujan conjecture exponent
We define C_56 = δ_2 to be the smallest real number δ ≥ 0 such that the following uniform bound toward the Generalized Ramanujan Conjecture holds.
0claims
0verified
Golay's Merit Factor
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
0claims
0verified
Good asymptotic constructions of Szemerédi–Trotter
If n,m are natural numbers, let C(n,m) denote the maximum number of incidences that are possible between n points and m lines in the plane. Establish upper and lower bounds on C(n,m) that are as strong as possible.
0claims
0verified
Gradient Descent Exponent
Let f be a convex function with 1-Lipschitz gradient. We assume black-box access to the function and its gradient. Gradient descent will converge to a global minimum with an appropriate choice of _step size_ s: x_k+1 := x_k - s· ∇ f(x_k). In general, s can be chosen to vary with the step k.
0claims
0verified
Hadwiger covering / illumination number in ℝ^3
C_39=H_3 is the Hadwiger covering number in dimension 3, which can also be formulated in terms of illumination of the boundary.
0claims
0verified
Hardy-Littlewood Maximal Inequality
Let C denote the best constant for which | x: sup_h>0 1/2h ∫_x-h^x+h f(y) dy ≥ λ | ≤ C/λ ∫_ℝ f(x) dx for absolutely integrable non-negative f : ℝ → ℝ. What is C?
0claims
0verified
Hausdorff-Young Inequality
For 1 ≤ p ≤ 2, let C(p) be the best constant such that ‖ hat f ‖_L^p'(ℝ) ≤ C(p) ‖ f ‖_L^p(ℝ) holds for all test functions f : ℝ → ℝ. Here p' := p/p-1 is the dual exponent of p. What is C(p)?
0claims
0verified
Heilbronn problem in a fixed bounding box
For any n ≥ 3 and any convex body K in the plane, let C(n,K) be the largest quantity such that in every configuration of n points in K, there exists a triple of points determining a triangle of area at most C(n,K) times the area of K. Establish upper and lower bounds on C(n,K).
0claims
0verified
Heilbronn problem in an arbitrary convex bounding box
For any n ≥ 3 let C(n) be the largest quantity such that in every configuration of n points in the plane, there exists a triple of points determining a triangle of area at most C(n) times the area of their convex hull. Establish upper and lower bounds on C(n).
0claims
0verified
How much can an autocorrelation of a non-negative function resemble an indicator function?
Let C be the best constant for which one has ‖f f‖_L^2(ℝ)^2 ≤ C ‖ff‖_L^1(ℝ) ‖f * f‖_L^∞(ℝ) for non-negative f : ℝ → ℝ. What is C?
0claims
0verified
Ihara constant over 𝔽_2
C_33=A(2) is the Ihara constant over 𝔽_2. <a href="#DM2013-def-Aq">[DM2013-def-Aq]</a> For each integer g≥ 1, let N_2(g) := maxbigl\#X(𝔽_2) : X/𝔽_2 a smooth projective geometrically integral curve of genus gbigr. <a href="#DM2013-def-Nqg">[DM2013-def-Nqg]</a> Then A(2) := limsup_g→inftyN_2(g)/g.
0claims
0verified
Ising perceptron capacity threshold
Let G = (g_ij) be an M × N random matrix with independent standard Gaussian entries, and let Z(G) := | σ ∈ -1,1^N : G σ ≥ 0 coordinatewise |. This is the zero-margin binary (or Ising) perceptron. Write M = ⌊ α N ⌋. Define C_80 to be the infimum of all α > 0 such that ℙ(Z(G) > 0) → 0 as N → ∞.
0claims
0verified
Kakeya and Nikodym sets in finite fields
Let d ≥ 1, and let q be a prime power. Let 𝔽_q be a finite field of order q. A Kakeya set is a set K that contains a line in every direction, and an Nikodym set N is a set with the property that every point x in 𝔽_q^d is contained in a line that is contained in N ∪ x.
0claims
0verified
Kakeya needle problem
Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…
0claims
0verified
Kakeya-type sum-difference constant
C_3b = SD(\0,1,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|)^C_3b whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± B := a ± b: a ∈ A, b ∈ B.
0claims
0verified
Komlós discrepancy constant
C_24 is the Komlós discrepancy constant (often denoted K). For a real matrix A∈ℝ^m× n, define its (sign) discrepancy by disc(A) := min_x∈-1,1^n ‖Ax‖_∞. For each n≥ 1, define the dimension-n Komlós discrepancy K_n := supdisc(A): A∈ℝ^n× n and ‖A_ast j‖_2≤ 1 for all columns j.
0claims
0verified
Korenblum's constant
Let D:=\z∈ℂ:lvert zrvert<1\. The Bergman space A^2(D) consists of analytic functions f on D with lVert frVert_2 := (1/π∫_D lvert f(z)rvert^2 dA(z))^1/2 < ∞, where dA(z) denotes the Lebesgue area measure. For c∈(0,1), write A(c,1) := z∈ℂ: c<lvert zrvert<1.
0claims
0verified
Landau's constant
Let D=\z∈ℂ:lvert zrvert<1\ and let F be the class of holomorphic functions f:D→ℂ normalized by f'(0)=1. <a href="#BS2023-def-F">[BS2023-def-F]</a> For finF, let L_f denote the radius of the largest disk contained in f(D).
0claims
0verified
Lebesgue universal covering constant
C_13b = a is the infimal area of a convex planar set Ω that can cover a congruent copy of every convex planar set of diameter 1.
0claims
0verified
Lehmer’s Mahler measure constant
Let f(x)=Σ_i=0^n a_i x^i = a_nΠ_i=1^n (x-α_i) be a polynomial with complex coefficients. The Mahler measure of f is M(f) := |a_n|Π_i=1^n max1,|α_i|.
0claims
0verified
Lindelof (pointwise growth) exponent for the Riemann zeta function
Define the infimal exponent μ_ζ by μ_ζ := infBiglθ≥ 0: lvertζ(1/2+it)rvert≪_ε(1+lvert trvert)^θ+ε for all ε>0Bigr. We define C_62a := μ_ζ, the Lindelof (pointwise growth) exponent for ζ(1/2+it).
0claims
0verified
Linnik's constant
For integers q≥ 2 and a with gcd(a,q)=1, let P(a,q) denote the least prime in the arithmetic progression a bmod q. <a href="#Xyl2011-def-Paq">[Xyl2011-def-Paq]</a> Linnik's theorem asserts that there exist constants C,L>0 such that P(a,q) ≤ C q^L (gcd(a,q)=1), uniformly for all q≥ 2.
0claims
0verified
Mahler volume product constant
Let K⊂ℝ^n be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying K=-K. Its polar body is K^∘ := y∈ℝ^n: ⟨ x,y⟩ ≤ 1 for all x∈ K. The volume product of K is vp(K) := Vol_n(K) Vol_n(K^∘).
0claims
0verified
Martinet's constant for totally real number fields
For a number field K, let Δ_K denote the absolute value of its discriminant and let [K:ℚ] denote its degree. The root discriminant of K is rd(K) := Δ_K^1/[K:ℚ].
0claims
0verified
Marton's conjecture (Polynomial Freiman-Ruzsa) constant
C_18 is the least constant such that, whenever A is a subset of 𝔽_2^n with lvert A+Arvert ≤ Klvert Arvert, then A can be covered by K^C_18+o(1) cosets of a subspace of cardinality at most lvert Arvert, where the limit o(1) is with respect to the limit K → ∞.
0claims
0verified
Matrix multiplications and AM-GM inequalities
For positive-semidefinite d × d matrices A_1, …, A_n and any unitarily invariant norm |||·||| (including the operator norm and Schatten p-norms) and m ≤ n, define C(n,m,d) := inf frac 1/n^m Σ_j_1, j_2, …, j_m = 1^n |||A_j_1A_j_2… A_j_m||| (n-m)!/n! Σ_substackj_1, j_2, …, j_m = 1 \ all distinct^n…
0claims
0verified
Max to min ratios
Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).
0claims
0verified
Maximal number of relevant variables in degree-d Boolean functions
Let f:0,1^n→0,1 be a Boolean function. Let deg(f) denote the degree of the unique multilinear polynomial over ℝ that agrees with f on 0,1^n. A variable x_i is relevant if f depends on it (equivalently: x_i appears in some monomial with nonzero coefficient in the multilinear representation of f).
0claims
0verified
Maximum Chromatic Number of Biplanar Graphs
C_27b is the highest possible chromatic number for any biplanar graph.
0claims
0verified
Metric TSP subtour-LP integrality-gap constant
In the symmetric metric traveling salesman problem, one is given a complete graph K_n=(V,E) with a nonnegative symmetric cost function c:E→ ℝ_≥ 0 satisfying the triangle inequality. For S⊆ V, let δ(S) denote the set of edges with exactly one endpoint in S, and write δ(v):=δ(\v\).
0claims
0verified
Minimal triangle density in graphs
For 0 ≤ ρ ≤ 1, let C(ρ) denote the largest quantity such that any graph on n vertices and (ρ+o(1)) C(n, 2) edges will have at least (C(ρ)-o(1)) C(n, 3) triangles. What is C(ρ)?
0claims
0verified
Moser's convex worm cover constant
C_13a is the infimal area of a convex domain Ω that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 1.
0claims
0verified
Moving Sofa Constant
The moving sofa constant C_41a=A is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle.
0claims
0verified
Multilinear Bohnenblust–Hille constant (real)
For integers m,n≥ 1, let B_ℝ,m(n) be the smallest constant such that every m-linear form T:(ℓ_∞^n)^m → ℝ satisfies the (multilinear) Bohnenblust–Hille inequality (Σ_j_1,…,j_m=1^n bigl|T(e_j_1,…,e_j_m)bigr|^2m/m+1)^m+1/2m ≤ B_ℝ,m(n) ‖T‖, where ‖T‖:=sup_‖x^(1)‖_∞,…,‖x^(m)‖_∞ ≤…
0claims
0verified
One-dimensional convex sub-Gaussian comparison constant
Let X be an integrable real random variable. We say that X is 1-sub-Gaussian in the tail sense if E[X]=0 quadand ℙ(lvert Xrvert>t)≤ 2e^-t^2/2quadfor all t≥ 0.
0claims
0verified
Packing in a dilate
For any n ≥ 1 and a geometric shape P (e.g. a polygon, a polytope or a sphere), let C(n, P) denote the smallest scale s such that one can place n identical copies of P with disjoint interiors inside another copy of P scaled up by a factor of s.
0claims
0verified
Pairwise touching cylinders
Is it possible for seven infinite circular cylinders C_1,…,C_7 of unit radius to touch all the others?
0claims
0verified
Points on sphere maximizing the volume
For any n ≥ 4, Let C(n) denote the maximum volume of a polyhedron with n vertices that all lie on the unit sphere S^2. What is C(n)? Which polyhedra attain the maximum volume?
0claims
0verified
Polya-Vinogradov best constant (squarefree asymptotic)
Let χ be a primitive Dirichlet character modulo q, and define S(χ) := max_N≤ q lvertΣ_1≤ n≤ Nχ(n)rvert. The Polya-Vinogradov inequality states that S(χ) ≤ c √(q) log q for some absolute constant c. <a href="#BK2020-def-PV">[BK2020-def-PV]</a> For squarefree moduli, define C_72^even (resp.
0claims
0verified
Rate at which κ(n) approaches 1
Given a real matrix A, let its condition number be κ(A):=σ_max(A)/σ_min(A), where σ_min(A) and σ_max(A) denote the smallest and largest singular values of A, respectively (with κ(A)=∞ if σ_min(A)=0).
0claims
0verified
Restriction exponent for the 2-sphere (Stein's L^∞ extension problem)
C_46 is the infimal exponent p such that one has the global bound ‖widehatf dσ‖_L^p(ℝ^3) lesssim_p ‖f‖_L^∞(S^2) qquadfor all f∈ L^∞(S^2).
0claims
0verified
Reverse Brunn-Minkowski constant
For subsets K,L⊂ℝ^n, their Minkowski sum is K+L := x+y: x∈ K, y∈ L. In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant.
0claims
0verified
Romanoff's constant
C_45 is the asymptotic density (if it exists) of the set of odd integers that can be expressed as the sum of a prime number and a power of two.
0claims
0verified
Rudin problem for polynomials
Let d ≥ 2 and D ≥ 1. For p ∈ 4,∞, let C^p(d,D) be the maximum of the ratio frac‖u‖_L^p(S^d)‖u‖_L^2(S^d) where u ranges over (real) spherical harmonics of degree D on the d-dimensional sphere S^d, which we normalize to have unit measure.
0claims
0verified
Schmeisser's Conjecture
For each n ≥ 2, let C(n) be the smallest constant such that for any complex polynomial f of degree n ≥ 2 with zeros z_1, …, z_n in the unit disk and critical points w_1, …, w_n-1, and for any nonnegative weights l_1, …, l_n ≥ 0 satisfying Σ_k=1^n l_k = 1, we have min_1 ≤ j ≤ n-1 | Σ_k=1^n l_k z_k -…
0claims
0verified
Schur–Siegel–Smyth trace constant
An algebraic integer α of degree d, with conjugates α_1,…,α_d, is totally positive if all of its conjugates are real and strictly positive. Its absolute trace (or trace-to-degree ratio) is overlinetr(α) := tr(α)/deg(α) = 1/dΣ_i=1^d α_i . Let A denote the set of totally positive algebraic integers.
0claims
0verified
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics