Scientists / Open questions

Jean Bourgain

Harmonic analysis; ergodic theory; partial differential equations

1994 Fields Medal

25 questions

Sharp $L^p$ bounds for Laplace eigenfunctions on the square torus (Bourgain's discrete restriction conjecture)
Sharp (m, s) tradeoffs for the sparse Johnson-Lindenstrauss transform on general sets: epsilon-dependence of the column sparsity, logarithmic losses, and biregular random models
Tensor-stability of the Radon–Nikodym property for projective tensor products of Banach spaces: the characterization problem left open by the Bourgain–Pisier counterexample
The optimal region for uniform $L^p$ resolvent estimates on compact Riemannian manifolds and the sharp spectral-projection scale on the flat torus
Improving the level of distribution of thin Pythagorean hypotenuse sets beyond 7/24 (Bourgain–Kontorovich, Beyond Expansion I)
Uniform $L^2$ restriction of toral eigenfunctions to hypersurfaces in dimensions $d \ge 4$ (Bourgain–Rudnick conjecture)
Nontrivial solutions of the shifted product equation $(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)$ for algebraic shifts: existence, optimal exponent, and uniformity
The cost of the equal-length constraint in approximating the Euclidean ball by zonotopes in ambient dimensions 3 and 4
Nodal intersections of toral eigenfunctions with a curved segment: the Bourgain-Rudnick linear lower-bound conjecture
The Farrell–Vershynin conjecture: optimal $O(\sqrt n)$ smoothed bound for $\|(D+R)^{-1}\|$ with bounded-density symmetric noise
Extending Bourgain–Klein's log-Hölder bounds on the density of states outer-measure to continuum Schrödinger operators in dimensions $d\ge 4$
Spectral gaps for dense subgroups of compact simple Lie groups beyond algebraic generators: the almost-Diophantine conjecture, quantitative bounds, and the semisimple case
Does the Kreĭn–Milman property imply the Radon–Nikodym property for arbitrary Banach spaces?
Geometric classification of the $\ell^2(L^6)$ decoupling for nondegenerate surfaces in $\mathbb{R}^4$
Sufficiency of Bourgain's bounded-entropy criterion for pointwise convergence of averages of commuting positive isometries
Minimal lifting index and loss of regularity for $S^1$ -valued Sobolev maps in dimension $n\ge3$ , and lifting in Bessel potential spaces
Effective exponents in composite-modulus sum-product estimates and explicit bounds for Heilbronn exponential sums
Size threshold for sum-free multiplicative subgroups of finite fields: closing the Alon–Bourgain gap between order $p^{1/3}$ and order $q^{3/4}$
Dimension-free $L^p$ bounds for the centered Hardy–Littlewood maximal operator over arbitrary centrally symmetric convex bodies
Box analogue of the Bourgain–Chang multilinear Burgess bound: power saving for $\chi(\prod_{j=1}^n L_j(\mathbf{x}))$ over axis-parallel boxes of volume exceeding $p^{n/4+\varepsilon}$
Sharpening the Bourgain–Garaev–Konyagin–Shparlinski bounds for congruences with products of variables from short intervals: the admissible-range exponent, the subpolynomial factor, and the application thresholds
Improving the tenth-moment bound $N_{10}(\delta,\Delta)$ for two-scale exponential sums: reducing the $(\delta+\Delta)N^6$ term in Bourgain's inequality (4.18) and closing the Theorem 9 range gap between $N^{-33/18}$ and $N^{-11/7}$
Primes in thin orbits, Selberg-quality spectral gaps for thin congruence subgroups, and effective saturation numbers: the unresolved core of the Bourgain–Gamburd–Sarnak 'Sieving and expanders' program
Falconer's distance set conjecture: does $\dim_H E>d/2$ force the distance set of $E\subset\mathbb{R}^d$ to have positive Lebesgue measure?

Back to open problems

Sharp LpL^pLp bounds for Laplace eigenfunctions on the square torus (Bourgain's discrete restriction conjecture)

Sharp (m, s) tradeoffs for the sparse Johnson-Lindenstrauss transform on general sets: epsilon-dependence of the column sparsity, logarithmic losses, and biregular random models

Tensor-stability of the Radon–Nikodym property for projective tensor products of Banach spaces: the characterization problem left open by the Bourgain–Pisier counterexample

The optimal region for uniform LpL^pLp resolvent estimates on compact Riemannian manifolds and the sharp spectral-projection scale on the flat torus

Improving the level of distribution of thin Pythagorean hypotenuse sets beyond 7/24 (Bourgain–Kontorovich, Beyond Expansion I)

Uniform L2L^2L2 restriction of toral eigenfunctions to hypersurfaces in dimensions d≥4d \ge 4d≥4 (Bourgain–Rudnick conjecture)

Nontrivial solutions of the shifted product equation (x1+σ)⋯(xν+σ)=(y1+σ)⋯(yν+σ)(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)(x1​+σ)⋯(xν​+σ)=(y1​+σ)⋯(yν​+σ) for algebraic shifts: existence, optimal exponent, and uniformity

The cost of the equal-length constraint in approximating the Euclidean ball by zonotopes in ambient dimensions 3 and 4

Nodal intersections of toral eigenfunctions with a curved segment: the Bourgain-Rudnick linear lower-bound conjecture

The Farrell–Vershynin conjecture: optimal O(n)O(\sqrt n)O(n​) smoothed bound for ∥(D+R)−1∥\|(D+R)^{-1}\|∥(D+R)−1∥ with bounded-density symmetric noise

Extending Bourgain–Klein's log-Hölder bounds on the density of states outer-measure to continuum Schrödinger operators in dimensions d≥4d\ge 4d≥4

Spectral gaps for dense subgroups of compact simple Lie groups beyond algebraic generators: the almost-Diophantine conjecture, quantitative bounds, and the semisimple case

Does the Kreĭn–Milman property imply the Radon–Nikodym property for arbitrary Banach spaces?

Geometric classification of the ℓ2(L6)\ell^2(L^6)ℓ2(L6) decoupling for nondegenerate surfaces in R4\mathbb{R}^4R4

Sufficiency of Bourgain's bounded-entropy criterion for pointwise convergence of averages of commuting positive isometries

Minimal lifting index and loss of regularity for S1S^1S1-valued Sobolev maps in dimension n≥3n\ge3n≥3, and lifting in Bessel potential spaces

Effective exponents in composite-modulus sum-product estimates and explicit bounds for Heilbronn exponential sums

Size threshold for sum-free multiplicative subgroups of finite fields: closing the Alon–Bourgain gap between order p1/3p^{1/3}p1/3 and order q3/4q^{3/4}q3/4

Dimension-free LpL^pLp bounds for the centered Hardy–Littlewood maximal operator over arbitrary centrally symmetric convex bodies

Box analogue of the Bourgain–Chang multilinear Burgess bound: power saving for χ(∏j=1nLj(x))\chi(\prod_{j=1}^n L_j(\mathbf{x}))χ(∏j=1n​Lj​(x)) over axis-parallel boxes of volume exceeding pn/4+εp^{n/4+\varepsilon}pn/4+ε

Sharpening the Bourgain–Garaev–Konyagin–Shparlinski bounds for congruences with products of variables from short intervals: the admissible-range exponent, the subpolynomial factor, and the application thresholds

Improving the tenth-moment bound N10(δ,Δ)N_{10}(\delta,\Delta)N10​(δ,Δ) for two-scale exponential sums: reducing the (δ+Δ)N6(\delta+\Delta)N^6(δ+Δ)N6 term in Bourgain's inequality (4.18) and closing the Theorem 9 range gap between N−33/18N^{-33/18}N−33/18 and N−11/7N^{-11/7}N−11/7

Primes in thin orbits, Selberg-quality spectral gaps for thin congruence subgroups, and effective saturation numbers: the unresolved core of the Bourgain–Gamburd–Sarnak 'Sieving and expanders' program

Falconer's distance set conjecture: does dim⁡HE>d/2\dim_H E>d/2dimH​E>d/2 force the distance set of E⊂RdE\subset\mathbb{R}^dE⊂Rd to have positive Lebesgue measure?

Sharp $L^p$ bounds for Laplace eigenfunctions on the square torus (Bourgain's discrete restriction conjecture)

The optimal region for uniform $L^p$ resolvent estimates on compact Riemannian manifolds and the sharp spectral-projection scale on the flat torus

Uniform $L^2$ restriction of toral eigenfunctions to hypersurfaces in dimensions $d \ge 4$ (Bourgain–Rudnick conjecture)

Nontrivial solutions of the shifted product equation $(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)$ for algebraic shifts: existence, optimal exponent, and uniformity

The Farrell–Vershynin conjecture: optimal $O(\sqrt n)$ smoothed bound for $\|(D+R)^{-1}\|$ with bounded-density symmetric noise

Extending Bourgain–Klein's log-Hölder bounds on the density of states outer-measure to continuum Schrödinger operators in dimensions $d\ge 4$

Geometric classification of the $\ell^2(L^6)$ decoupling for nondegenerate surfaces in $\mathbb{R}^4$

Minimal lifting index and loss of regularity for $S^1$ -valued Sobolev maps in dimension $n\ge3$ , and lifting in Bessel potential spaces

Size threshold for sum-free multiplicative subgroups of finite fields: closing the Alon–Bourgain gap between order $p^{1/3}$ and order $q^{3/4}$

Dimension-free $L^p$ bounds for the centered Hardy–Littlewood maximal operator over arbitrary centrally symmetric convex bodies

Box analogue of the Bourgain–Chang multilinear Burgess bound: power saving for $\chi(\prod_{j=1}^n L_j(\mathbf{x}))$ over axis-parallel boxes of volume exceeding $p^{n/4+\varepsilon}$

Improving the tenth-moment bound $N_{10}(\delta,\Delta)$ for two-scale exponential sums: reducing the $(\delta+\Delta)N^6$ term in Bourgain's inequality (4.18) and closing the Theorem 9 range gap between $N^{-33/18}$ and $N^{-11/7}$

Falconer's distance set conjecture: does $\dim_H E>d/2$ force the distance set of $E\subset\mathbb{R}^d$ to have positive Lebesgue measure?