arXiv subjects

David Victor Feldman

Publications and source records attributed to David Victor Feldman.

5 recordsLinked to original sources

arXiv2026

Elementary equivalence of convex bodies in affine and projective languages

Convex subsets of R^n carry two first-order structures: barycentric (affine) structure, with operations C_l(p,q)=(1-l)p+lq for l in [0,1], and betweenness (projective) structure, with ternary relation B(a,x,b) meaning x lies in [a,b]. Isomorphism means affine equivalence in the first and, for n at least 2 and sets open or closed, projective equivalence in the second. We ask when elementary equivalence already determines the body. Main theorem: two compact convex bodies of any dimension, with no regularity hypotheses, are elementarily equivalent in the barycentric language if and only if they are affinely equivalent. The proof rests on a definable compact family of gauges: simplices stationary for barycentric coordinates, with volume bounded below via the anticomplementary simplex. For the betweenness language we develop an interior von Staudt calculus, all quantifiers ranging over the body, making harmonic conjugacy and rational cross-ratio comparisons first-order; a relativization scheme then propagates planar separations to R^n. Consequences: the closed unit ball is separated from sum x_i^4 <= 1 for every n at least 2; and in the plane, projective categoricity holds outright for convex polygons and for bodies with real-analytic, positively curved, non-conic boundary, the latter via a definable finite projective invariant, the conic-cluster set: the points whose every boundary arc contains six co-conic extreme points. A general reduction isolates what remains of the projective conjecture: recovery of boundary coordinates in dimension at least three, and a definable compact gauge, obstructed exactly by non-compact projective symmetry, as on the quadric. For closed noncompact bodies the asymptotic structure is itself elementary: the dimensions of the recession cone and of the lineality space are determined by the betweenness theory, separating the solid cylinder from the slab.

math.MG↗

arXiv2026

Divisors in windows and the Euler--Mascheroni constant

For $0<c<1$ the number of divisors of $q$ below $q^{c}$ averages, over $q\le B$, to $c\log B+(γ-c)+O(B^{-\min(c,1-c)})$, where $γ$ denotes the Euler--Mascheroni constant. Consequently the number of divisors in a window $(q^{a},q^{b})$ averages to $(b-a)(\log B-1)+o(1)$: the constant $γ$ cancels, and the average counts for two windows stand asymptotically in the ratio of the windows' exponent lengths. In particular a number has, on average, twice as many divisors in $(n^{1/4},n^{1/2})$ as in $(n^{1/8},n^{1/4})$, by an argument nowhere mentioning $γ$. Anchoring a window at a threshold and its square root makes the logarithms cancel instead: for weakly increasing $F$ with $F(q)\to\infty$ and $q/F(q)\to\infty$, a number $q$ has, on average, exactly $γ$ more divisors below $\sqrt{F(q)}$ than in $[\sqrt{F(q)},F(q))$. For windows on the scale of $q$ itself the average depends on an endpoint convention: with $α\in(0,1)$, counting divisors below $\sqrt{αq}$ against divisors in $(\sqrt{αq},αq)$ yields the average $H_{\lfloor 1/α\rfloor}-\log(1/α)$, while the window $(\sqrt{αq},αq]$ yields $H_{\lceil 1/α\rceil-1}-\log(1/α)$; the two constants differ exactly when $1/α$ takes an integer value, and then by $α$. Either function of $α$ oscillates about $γ$, tends to $γ$ as $α\to0$, and integrates over $(0,1)$ to $ζ(2)-1$. The principal results are formally verified in Lean~4.

math.NT↗

arXiv2026

Arranging convex bodies for maximum intersection volume: the sharp efficiency of centroid alignment

Given finitely many compact convex bodies in $\R^n$, one seeks translates maximizing the volume of their common intersection. A lazy solver merely translates each body so as to place its centroid at the origin. We prove that the lazy strategy always captures strictly more than $\left(\tfrac{2}{n+1}\right)^n$ of the optimal volume, and that this constant is sharp: families of cones over tangent disks, indexed by finite nets on the sphere, approach it. The infimum is not attained. For two convex bodies in the plane the resulting sharp constant $4/9$ closes a gap open since 1996, when de Berg, Cheong, Devillers, van Kreveld and Teillaud proved that centroid alignment of two convex polygons captures at least $9/25$ of the maximum overlap and exhibited examples capturing only $4/9$. The proof rests on the following identity: for a convex body $K$ with centroid at the origin, the intersection of all centroid-recentered compact convex supersets of $K$ equals $\tfrac{1}{n+1}(K-K)$. We close with a promise-problem variant in which the lazy strategy captures at least $\left(\tfrac{n}{n+1}\right)^n > \tfrac1e$ of the optimum, uniformly in the dimension. All results below have been checked in Lean~4; the one classical input quoted rather than proved is the equality case of the Brunn--Minkowski inequality, which enters only for $n\ge2$.

math.MG↗

arXiv2026

Plücker coordinates of finite-dimensional subspaces of $\ell^p$ and its direct sums: summability, reconstruction, stratification

An $n$-dimensional subspace of $\ell^p$ has Plücker coordinates indexed by the $n$-element subsets of $\N$. We show these coordinates lie in $\ell^p\In{n}$ --- the exponent is preserved --- with multilinear norm exactly $1$ for $0 2$ the sharp constant exceeds $1$ and its determination contains the Hadamard maximal determinant problem. A reconstruction lemma shows every nonzero solution of the quadratic Plücker relations in $\ell^p\In{n}$ is decomposable with frame in $\ell^p$; consequently $\Gr_n(\ell^p)$ is a closed Banach-analytic submanifold of $\mathbb{P}\big(\ell^p\In{n}\big)$ cut out by the Plücker relations alone, with no auxiliary summability condition and no polarization. For mixed sums $\bigoplus \ell^{p_i}$ the exterior power is graded by compositions of $n$; the support of the grading is the lattice-point set of a generalized permutohedron determined by the intersection pattern of the subspace with partial sums, this stratification is canonical for the isometry group though not for $\GL$, and each stratum admits a tubular neighborhood whose normal coordinates are precisely the Plücker blocks vanishing on it. We record what is proved and what is conjectured; the finitary and single-space core of the theory, including full proofs of Cauchy--Binet and Hadamard's inequality, has been formally verified in Lean~4.

math.FA↗

arXiv2026

Anomalous Partial Quotients in the Continued Fraction of $\sqrt{ζ(3)-S_N}$

Let $S_N = \sum_{j=1}^N j^{-3}$ and $R_N = ζ(3) - S_N$. The simple continued fraction of $\sqrt{R_N}$ has partial quotients of generic size $O(N)$. We prove that at the sequence of indices $N_k = (Q_{2k+1}-1)/2$, where $Q_{2k+1}$ are companion Pell numbers, the continued fraction begins \[ \sqrt{R_{N_k}} = \bigl[0;\; M_k-1,\; 1,\; 6M_k^3+12M_k-2,\; 1,\; \ldots\,\bigr], \] with $M_k = P_{2k+1}$ (Pell numbers), and the third partial quotient grows cubically while generic ones are linear. We determine all partial quotients through the fifth: \begin{align*} \PQ_0 &= M_k - 1, & \PQ_2 &= 6M_k^3 + 12M_k - 2, & \PQ_4 &= \Bigl\lfloor\frac{10M_k - 261}{261}\Bigr\rfloor, \PQ_1 &= 1, & \PQ_3 &= 1, & \PQ_5 &= \Bigl\lfloor\frac{261}{r_k}\Bigr\rfloor + ε_k, \end{align*} where $r_k = (10M_k) \bmod 261$ satisfies the recurrence $r_{k+1} \equiv 6r_k - r_{k-1} \pmod{261}$, and $ε_k = -1$ at the $k$ with $r_k \mid 261$ (the two residue classes $k \equiv 57, 62 \pmod{60}$), and $ε_k = 0$ otherwise. All six formulas follow from the Euler--Maclaurin expansion of $1/\sqrt{R_{N_k}}$, carried to sufficient precision, combined with the Pell identity $Q_{2k+1}^2 - 2M_k^2 = -1$. The delicate first step, $\PQ_0 = M_k - 1$, is proved by rationalizing the irrational factor $\sqrt{2}$ in the Euler--Maclaurin expansion; we complement this proof with a heuristic derivation via Gosper's bihomographic continued-fraction algorithm that exposes the underlying mechanism. All claimed results have been formalized in LEAN with the aid of Aristotle.

math.NT↗