An autocorrelation problem related to difference bases
Let C be the smallest constant such that min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C ‖f‖_L^1(ℝ)^2 for f ∈ L^1(ℝ). What is C?
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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.
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An autocorrelation problem with functions that can take negative values
Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). What is C?
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Approximation ratio for quantum Max Cut
Quantum Max Cut is the quantum analog of Max Cut. Given a graph G = (E,V), it asks for the maximum eigenvalue of H_G = Σ_(ij) ∈ E (I - X_iX_j - Y_iY_j - Z_i Z_j), where X_i, Y_i, Z_i are the Pauli matrices acting on the i'th tensor factor and trivially on all other coordinates.
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Asymptotic counting exponent for partial Hadamard matrices
For integers n ≥ 2 and t ≥ 1, an n × t partial Hadamard matrix is a matrix with entries in \± 1\ whose rows are pairwise orthogonal. Let N_n,t denote the number of such matrices. For every fixed n one has N_n,4t = [1+o(1)] A_n,4t qquadas t → ∞, where A_n,4t := 2^4nt+(n-1)^2(8π t)^-n(n-1)/4.
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Asymptotic Dobrowolski constant for Lehmer’s problem
Let α be a nonzero algebraic number of degree d, with minimal polynomial over ℤ f(X)=a_dΠ_i=1^d (X-α_i), where a_d>0 and α_1,…,α_d are the conjugates of α. Define the Mahler measure of α by M(α) := a_dΠ_i=1^d max1,lvert α_irvert.
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Asymptotic essential-dimension ratio of the symmetric groups
For each integer n ≥ 1, let S_n be the symmetric group on n letters. Over a base field k, the essential dimension ed_k(S_n) is the smallest integer d such that the general degree-n polynomial x^n + a_1 x^n-1 + ⋯ + a_n can be reduced to a d-parameter form by a Tschirnhaus transformation.
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Asymptotic line-count constant for smooth degree-d surfaces in ℙ^3 in characteristic 0
For each integer d ≥ 3, let ℓ_0(d) denote the maximal number of lines contained in a smooth surface of degree d in ℙ^3_ℂ. We define C_76 := limsup_d→∞ℓ_0(d)/d^2. The constant C_76 measures the quadratic growth rate of the maximal line count on smooth complex degree-d surfaces.
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Beurling–Ahlfors transform constant
In harmonic analysis, the Beurling–Ahlfors transform B (also called the Ahlfors–Beurling operator) is the singular integral operator on L^p(ℂ), 1<p<∞, defined by Bf(z) = -1/π p.v.∫_ℂ f(w)/(z-w)^2 dm(w) = -1/π lim_ε→ 0^+∫_lvert w-zrvert>varepsilonf(w)/(z-w)^2 dm(w), where dm is Lebesgue measure on…
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Bloch’s constant
Let D=\z∈ℂ:lvert zrvert<1\. Following standard notation, let F be the class of holomorphic functions f:D→ℂ normalized by lvert f'(0)rvert=1 (equivalently, after rotation, f'(0)=1).
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Block Stacking Problem
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
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Bohnenblust–Hille constant on the Boolean cube
Degree at most d functions f:lbrace ± 1rbrace^n→ℝ have Fourier–Walsh expansion f(x)=Σ_S⊆ [n], |S|≤ d widehat f(S) x^S, x^S:=Π_i∈ Sx_i, [n]:=lbrace 1,…,nrbrace. For d∈ℕ set p_d:=2d/d+1.
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Bohr radius for the bidisc
Let D^d := z=(z_1,…,z_d)∈ℂ^d: lvert z_1rvert,…,lvert z_drvert<1 be the unit polydisc, and let the Schur class S_d be the set of analytic functions f:D^dtoD.
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Borcea's Conjecture
For any 1 ≤ p < ∞ and n ≥ 2, let C(p,n) be the smallest constant such that for any complex polynomial f of degree n with zeroes z_1,…,z_n satisfying 1/n Σ_i=1^n |z_i|^p ≤ 1, and every zero f(ζ)=0 of f, there exists a critical point f'(ξ) = 0 of f with |ξ - ζ| ≤ C(p,n). What is C(p,n)?
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Bounded prime gap constant
Let p_n denote the n-th prime. The bounded prime gap constant is C_88a = H_1 := liminf_n → ∞ (p_n+1 - p_n), the least limit point of the sequence of gaps between consecutive primes.
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Brennan's conjecture exponent
Let Ω⊂ℂ be simply connected with at least two boundary points in the extended complex plane, and let φ:ΩtoD be a conformal map. Brennan's conjecture states that ∫_Ωlvert φ'(z)rvert^p dx dy < ∞ qquadwhenever 4/3<p<4.
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Brezis–Gallouet–Wainger remainder constant on the 2D torus
C_16 = L is the smallest constant for which the sharp Brezis–Gallouet inequality ‖u‖_L^∞(T^2)^2 ≤ 1/4π ‖∇ u‖_L^2(T^2)^2 Bigl[lnδ(u) + lnbigl(1+lnδ(u)bigr) + LBigr] holds for all zero-mean functions u ∈ H^2(T^2) with sufficiently large frequency ratio δ(u) := ‖Δ u‖_L^2(T^2)^2/‖∇ u‖_L^2(T^2)^2.
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Brun's Constant
C_81a, Brun's Constant, is the sum of the reciprocals of the twin primes.
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Burgess-quality subconvexity exponent for Dirichlet L-functions
A Dirichlet character of level q is an arithmetic function χ that is multiplicative, is defined by a character on (ℤ/qℤ)^ast on integers coprime to q, and is 0 on integers not coprime to q.
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Centered Hardy–Littlewood maximal constant in dimension 2
In ℝ^d (d≥ 1), let M_d denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by M_d f(x) := sup_r>0 1/lvert Q(x,r)rvert∫_Q(x,r) lvert f(y)rvert dy, where Q(x,r) is a closed ℓ_∞ ball of radius r and center x in ℝ^d, that is, a closed cube centered at x, with sides…
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Chvátal–Sankoff constant for a binary alphabet
Let λ_n,2 be the random variable assigning two uniformly random binary strings of length n the length of their longest common subsequence. Then C_31a is the (well-defined) limit C_31a := lim_n → inftyE[λ_n,2]/n.
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Classical zero-free region constant
C_8 = R is the least constant such that there are no zeroes σ+it of the Riemann zeta function with lvert t rvert ≥ 2 and σ > 1 - 1/R log lvert t rvert.
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Constant term of one-shot channel simulation
The constant term of one-shot channel simulation [HJMR07], [BG14], [LEG18], [Li25] is given as (we use the definition in [Li25]) C_32=limsup_t→∞(sup_p_X,Y: I(X;Y)=t inf_p_S|X,Y: I(X;S)=0H(Y|S)-t-log_2t), where H(Ylvert S)=H(Y,S)-H(S) is the conditional entropy (in bits), and I(X;Y)=H(X)+H(Y)-H(X,Y)…
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Conway thrackle constant
In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.
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Davenport constant for C_n^3
In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈ℕ such that every sequence S over G of length lvert Srvert≥ l has a non-empty zero-sum subsequence.
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de Bruin-Sharma Problem
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
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Difference Bases
For any natural number n, let Δ(n) be the size of the smallest set B of integers such that every natural number from 1 to n is expressible as a difference of two elements of B (such sets are known as difference bases for the interval 1,…,n). Write C(n) := Δ^2(n)/n, and C := inf_n ≥ 1 C(n).
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Dirichlet divisor problem exponent
Let d(n) be the divisor function. The Dirichlet divisor problem concerns the error term Δ(x) := Σ_n≤ x d(n) - x(log x + 2γ -1), where γ is Euler's constant. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a> Define the divisor-problem exponent α := infBigla≥ 0: Δ(x)=O(x^a+ε) for all ε>0Bigr.
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Dual matrix multiplication exponent
In algebraic complexity theory, for each real k ≥ 0, let ω(k) denote the exponent for multiplying an n × n^k matrix by an n^k × n matrix. We define α := supk ≥ 0 : ω(k) = 2.
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Elliott-Halberstam level-of-distribution exponent
Let Λ denote the von Mangoldt function. For coprime positive integers a,q, define ψ(x;q,a) := Σ_n≤ x, n≡ a (mod q) Λ(n).
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Equidistant points in convex polygons
Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?
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Erdős maximum-term constant
For any transcendental entire function f(z)=Σ_n≥ 0 a_n z^n, define . M(r,f):=max_|z|=r|f(z)|, μ(r,f):=max_n≥ 0|a_n| r^n. Following [Er1961], define β(f):=liminf_r→∞μ(r,f)/M(r,f). We define C_51 = B to be the supremum of β(f) over all transcendental entire functions f.
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Erdős squarefree problem
For any natural number N, let C(N) denote the largest cardinality of a subset A of 1,…,N with the property that ab+1 is square-free for all a,b ∈ A. Establish upper and lower bounds for C(N) that are as strong as possible.
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Erdős squares in a square problem
For any natural n, let C(n) denote the maximum possible sum of side lengths of n squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for C(n) that are as strong as possible.
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Erdős–Gyárfás conjecture
Let G be a finite graph with minimum degree at least 3. Must G contain a cycle of length 2^k for some k ≥ 2?
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Erdős–Szemerédi 3-sunflower-free capacity
A family of three distinct sets A,B,C is a 3-sunflower (or Δ-system) if A∩ B = A∩ C = B∩ C. A family of sets is sunflower-free if it contains no 3-sunflower (equivalently, no sunflower of any size ≥ 3). Let [n]:=\1,2,…,n\ and let f(n) denote the maximum size of a sunflower-free family F⊆ 2^[n].
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Essential minimum of the Zhang-Zagier height
Let overlineℚ be the set of all algebraic numbers. The naïve height h : overlineℚ → ℝ is defined as follows. Let α ∈ overlineℚ and let P(x) be an irreducible primitive polynomial with integers coefficients such that P(α)=0. Let n be the degree and a be the leading coefficient of P(x).
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Exponent for bounded gaps between many primes
Let p_n denote the n-th prime and, for m ≥ 1, write H_m := liminf_n → ∞ (p_n+m - p_n) for the least limit point of the gaps between primes m apart.
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Exponent for commutators close to the identity
Let H be an infinite-dimensional complex Hilbert space and let B(H) be the Banach algebra of bounded operators on H, equipped with the operator norm.
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Exponential growth constant for diagonal Ramsey numbers
C_17 is the limit (if it exists) of R(k)^1/k as k → ∞, where the diagonal Ramsey number R(k) is the smallest integer n such that every red/blue colouring of the edges of the complete graph K_n contains a monochromatic copy of K_k.
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Factoring N! into N numbers
For a natural number N, let C(N) be the largest quantity such that N! can be factored into N factors that are greater than or equal to C(N) (see OEIS A034258). Establish upper and lower bounds on C(N) that are as strong as possible.
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Falconer distance problem in ℝ^2
The Falconer distance problem threshold C_34 = s_Δ(ℝ^2) in the plane is defined as s_Δ(ℝ^2) : :=\ infBigl s∈[0,2] : ∀ compact E⊂ℝ^2,\ dim_H(E)>s Longrightarrow lvertΔ(E)rvert>0 Bigr.
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Favard-length decay exponent
Let E⊂ ℝ^2 be a planar set. The Favard length of E is defined by Fav(E) := 1/π∫_0^π lvert Proj R_θ Ervert dθ, where Proj is orthogonal projection to the horizontal axis and R_θ is rotation by angle θ.
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Flatness constant in dimension 3
A convex body K⊂ℝ^d is hollow (lattice-free) with respect to a lattice Λ if int(K)∩Λ=∅. <a href="#CS2019-hollow-def">[CS2019-hollow-def]</a> For a hollow body, the lattice width is w(K) := min_u∈ℤ^d∖0 (max_x∈ Ku· x-min_x∈ Ku· x).
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Fourier Entropy-Influence constant
Let f:\-1,1\^n→\-1,1\ be a Boolean function with Fourier expansion f(x)=Σ_S⊆[n]hat f(S)χ_S(x). Its spectral entropy is H(hat f^2) := Σ_S⊆[n]hat f(S)^2log_21/hat f(S)^2, and its total influence is Inf(f) := Σ_S⊆[n]hat f(S)^2 lvert Srvert.
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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).
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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.
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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).
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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…
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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.
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Golay's Merit Factor
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
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