Analysis

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E0120/: Asks whether, for every infinite set of reals, some set of positive measure contains no affine copy of it.

E0225/: Asks whether a trigonometric polynomial with all real roots and maximum modulus one has integral of its absolute value at most 4 over a full period.

E0226/: Asks whether some entire non-linear function sends exactly the rational real numbers to rational values.

E0227/: Asks whether, for a non-polynomial entire function, the limiting ratio of largest coefficient term to maximum modulus, when it exists, must be zero.

E0229/: Asks whether, given sets of complex numbers with no finite limit point, some transcendental entire function has each set among zeros of some derivative.

E0256/: Estimates the largest lower bound for the maximum modulus on the unit circle of a product of terms one minus z to the a-i, over all choices of n exponents.

E0395/: Asks whether random signs on n unit complex numbers give a sum of absolute value at most the square root of two with probability at least about one over n; Erdős asked it with radius one, which fails for every even n.

E0498/: Asks whether at most the middle binomial coefficient of the signed sums of n complex numbers of modulus at least one can lie in one open unit disc.

E0510/: Asks whether every set of N positive integers admits an angle where the sum of the cosines of its members times that angle is below a negative constant times root N; the site's wording over all integers fails at sets containing zero.

E0511/: Asks whether the set where a monic polynomial has modulus below one has boundedly many components of diameter above a fixed constant, whatever the degree.

E0512/: Asks whether the mean absolute value of the exponential sum over a set of N integers is at least of order the logarithm of N.

E0513/: Determines the largest limit inferior of the ratio of the maximal power series term of a transcendental entire function to its maximum modulus on radius r.

E0514/: Asks whether every transcendental entire function has a path to infinity on which it outgrows every power of the variable, how long such a path must be, and whether it can outgrow a fixed function of the maximum modulus.

E0515/: Asks whether every entire nonpolynomial function has a rectifiable path to infinity along which the integral of any negative power of its modulus is finite.

E0516/: Asks whether an entire function of finite order with very sparse exponents has minimum modulus whose logarithm matches that of its maximum modulus.

E0517/: Asks whether an entire power series whose exponents grow faster than linearly in the index must take every complex value infinitely often.

E0519/: Asks whether the largest modulus among the first n power sums of complex numbers, one of which is one, is bounded below by an absolute positive constant.

E0527/: Asks whether, for almost all sign choices, a signed power series with small but not square-summable coefficients converges somewhere on the unit circle.

E0671/: Concerns sequences of Lagrange interpolation polynomials built on nodes in the interval from minus one to one and how they behave as the degree grows.

E0906/: Asks whether there is a transcendental entire function whose derivatives, along any infinite subsequence of orders, have zero sets that together are dense in the plane.

E0907/: Asks whether a real function whose every fixed-shift difference is continuous must be the sum of a continuous function and an additive one.

E0908/: Asks whether a real function whose every fixed-shift difference is measurable splits into measurable, additive and almost-everywhere-shift-invariant parts; the site prints continuous.

E0909/: Asks whether, for each n at least two, there is a space of dimension n whose square also has dimension n.

E0910/: Asks whether every connected set in n-dimensional space has a connected subset that is neither a single point nor homeomorphic to the whole set.

E0967/: Asks whether, for integer sequences whose reciprocals sum finitely, one plus the sum of their reciprocals to the power one plus i t is never zero.

E0973/: Asks whether n complex numbers of modulus at least one, the first equal to one, can keep all their power sums below an exponentially small bound.

E0974/: Asks whether complex numbers whose power sums vanish on infinitely many blocks of n minus one consecutive indices are essentially the nth roots of unity, read as Tijdeman's classification.

E0990/: Asks whether a polynomial's root arguments are equidistributed with error at most the square root of the number of nonzero coefficients times a log factor.

E0996/: Asks how fast a square integrable function's Fourier partial sums must converge for its averages along alpha times a lacunary sequence to equal its integral.

E1038/: Determines the smallest and largest measure of the set where a monic real polynomial with all roots real in minus one to one has absolute value below one.

E1040/: The least area of the region where a monic polynomial with all roots from a fixed closed infinite set of complex numbers has absolute value below one.

E1042/: Asks what can be said about a closed set in the plane of transfinite diameter one that lies inside no closed disc of radius one.

E1043/: Asks whether every monic non-constant complex polynomial has a line onto which the set where its absolute value is at most one projects to measure at most two.

E1044/: Determines the infimum of the largest component boundary length of the region where a polynomial with all roots in the closed unit disc is below one.

E1045/: The maximum product of all pairwise distances among complex numbers that are pairwise at most distance two apart, and whether a regular polygon is optimal.

E1046/: Asks whether the set where a monic complex polynomial has absolute value below one must lie in a disc of radius two whenever that set is connected.

E1047/: Examines a monic polynomial with m distinct roots and a threshold so small that the set where its absolute value is at most that threshold has m components.

E1048/: Asks whether a monic polynomial with all roots of absolute value at most r below two has a component of diameter over two minus r where it is below one.

E1115/: The linear-length asymptotic-path conjecture fails at every finite order, even for functions growing arbitrarily close to the logarithmic-square threshold that guarantees radial paths.

E1116/: Asks whether some meromorphic or entire function has, for every two distinct values, the ratio of their solution counts in growing discs unbounded.

E1117/: Concerns the number of points on the circle of radius r at which a non-monomial entire function attains its maximum modulus.

E1118/: Concerns non-constant entire functions for which the set where the modulus exceeds some constant has finite measure.

E1120/: Asks for the shortest path from zero to the unit circle inside the set where a monic polynomial with all roots in the unit disc has modulus at most one.

E1125/: Asks whether a real function with twice its value at a point at most the sum of its values at two later equally spaced points must be monotonic.

E1126/: Asks whether a function additive for almost all pairs of reals must agree almost everywhere with a function that is additive for all pairs.

E1154/: Asks whether, for every number between zero and one, some ring or field of real numbers has exactly that Hausdorff dimension; yes under the continuum hypothesis by Mauldin's theorem, so the existence cannot be refuted in ZFC.

E1164/: Asks whether the logarithm of the radius of the largest origin-centered disc a planar simple random walk covers by time n has order sqrt(log n) in probability; the site's almost-every wording is corrected, and Révész and Dembo–Peres–Rosen–Zeitouni prove it.

E1165/: Asks the infinitely-often probability of exactly r sites tied for maximum local time in planar simple random walk, for each integer r at least three.

E1166/: Asks whether the number of sites ever favorite by time n in planar simple random walk is eventually bounded by a power of log n almost surely.

E1195/: Concerns sets of real numbers of infinite measure in which no ratio of two distinct elements is an integer.

E1197/: Asks whether, for a set of positive measure and almost every positive x, every large integer multiple of x lies in some integer dilate of the set.

E1215/: Asks whether a constant bounds the length of a path from zero to the unit circle inside the region where such a polynomial has modulus below one.

E1221/: Asks whether, for a sequence on the circle, the de Bruijn–Erdős constants for the largest and smallest sums of r consecutive gaps and for their ratio deviate from their trivial values by more than any constant over r as r grows; a 2026 preprint claims all three parts for distinct points, unrefereed and unreviewed.

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Entire and complex functions, real functions and measure, exponential sums and series, and the few purely probabilistic and topological problems that carry no subject co-tag.

Site tags routed here: analysis, combinatorics, iterated functions, number theory, probability, topology.