On volumes of simplices in intermediate dimensions

Eyvindur Ari Palsson; Jos\'e Gaitan Montejo

arxiv: 2605.22450 · v1 · pith:IQWYNBHYnew · submitted 2026-05-21 · 🧮 math.CA

On volumes of simplices in intermediate dimensions

Jos\'e Gaitan Montejo , Eyvindur Ari Palsson This is my paper

Pith reviewed 2026-05-22 01:30 UTC · model grok-4.3

classification 🧮 math.CA

keywords Hausdorff dimensionsimplex volumesFalconer distance problemBeck theoremMarstrand projection theoremincidence geometryBorel sets

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The pith

For ambient dimensions d up to 2k a Hausdorff dimension of k guarantees positive Lebesgue measure on (k+1)-simplex volumes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a variant of the Falconer distance problem that asks how large the Hausdorff dimension of a Borel set E in R^d must be to ensure that some k+1 points in E span a family of (k+1)-simplices whose volumes have positive Lebesgue measure. It extends the sharp threshold k previously known only for d equal to k+1 to the full range k+1 less than or equal to d less than or equal to 2k. For ambient dimensions larger than 2k the paper obtains the weaker but still nontrivial threshold d minus k. The argument transfers incidence and projection theorems to this geometric setting.

Core claim

If the Hausdorff dimension of a Borel set E subset R^d exceeds k when k+1 less than or equal to d less than or equal to 2k, or exceeds d minus k when d greater than 2k, then there exist points x0 through xk in E such that the (k+1)-volumes of the simplices formed by adding any further point x_{k+1} from E form a set of positive Lebesgue measure.

What carries the argument

Application of the continuum Beck-type theorem for hyperplanes together with Marstrand projection and slicing theorems to control the distribution of simplex volumes.

If this is right

When the ambient dimension is at most 2k the dimensional threshold remains k, matching the earlier result for the lowest dimension.
When the ambient dimension exceeds 2k the threshold rises continuously to d minus k.
The same conclusion holds for any Borel set satisfying the dimension hypothesis.
An alternative elementary proof route exists under the additional Fubini property for Hausdorff dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The transition point at ambient dimension 2k suggests that the relation between configuration dimension k and ambient dimension d governs the sharpness of geometric incidence results.
The same incidence-projection strategy could be tested on other Falconer-type problems that involve volumes or higher-order distances in intermediate dimensions.
Explicit fractal constructions at the threshold dimension in the range 2k less than d less than 3k would clarify whether the bound d minus k is sharp.

Load-bearing premise

The incidence and projection theorems used in the proof transfer directly to the specific configuration of simplex volumes without additional restrictions.

What would settle it

A Borel set E in R^d with Hausdorff dimension exactly k (for d less than or equal to 2k) or exactly d minus k (for d greater than 2k) such that every choice of k+1 points produces a zero-measure set of simplex volumes with the remaining points in E would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.22450 by Eyvindur Ari Palsson, Jos\'e Gaitan Montejo.

**Figure 1.** Figure 1: Representation of B ⊂ R m+n sliced by π −1 X , in relation to the Fubini property. It is well known that the Fubini property (1.2) does not hold in general, for instance, Falconer in [5, 6] provides the construction of two classic examples. See also [1, Theorem 2.2] and references therein for a related example with f : [0, 1] → [0, 1]n whose graph has Hausdorff dimension n + 1. Example 1. [6, Example 7.8] … view at source ↗

**Figure 2.** Figure 2: In Figure (a) we represent how the size of the set |ProjW⊥(E)| dictates the amount of different volumes of k-simplices with a base contained in W. In Figure (b) the origin is O = ProjW ⊥ (W), and any parallel affine plane gets mapped to a point with distance d(O,ProjW ⊥ (y + W)) = r. Proof of Theorem 1.1. Let E be a Borel set such that dimH(E) > s > k and µ be a s-F rostman measure supported on E. By the d… view at source ↗

**Figure 3.** Figure 3: Graph of some fs with Hausdorff dimension 1 < s < 2. The set B from [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Fubini property along different directions. Notice that the first condition of Theorem 1.9, that demands the incidence set (1.7) to satisfy the Fubini property is equivalent to requiring the graph of ψk|Ek+1\D to have the Fubini property along the horizontal direction πY := Proj{0}m×Rn , which is a plausible situation, at least for a graph of a function fs as the one from [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 5.** Figure 5: Set E ⊂ R 3 with dE = 2, but any 1-F rostman measure supported on E intersection a 2-plane is not irreducible. References [1] H´era, K., Keleti, T., M´ath´e, A. A Fubini-type theorem for Hausdorff dimension. JAMA 152, 471–506 (2024). https://doi.org/10.1007/s11854-023-0302-3 [2] P. Bright, A. Ortiz, D. Zakharov, A Continuum Beck-type Theorem for Hyperplanes (2025), arXiv:2510.10907. [3] X. Du, Y. Ou, K. Re… view at source ↗

read the original abstract

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \leq d \leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of H\'{e}ra, Keleti and M\'{a}th\'{e}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the simplex volume threshold from d=k+1 into the intermediate range k+1 to 2k using Beck and Marstrand tools, but the nonlinear transfer needs explicit verification.

read the letter

The main point is that this paper takes the sharp threshold from Shmerkin and Yavicoli for when a set in R^{k+1} forces positive measure in (k+1)-simplex volumes and extends it to k+1 ≤ d ≤ 2k with the same threshold, plus a weaker d-k bound for d > 2k. That is the concrete advance over the base case they cite. They motivate it from the earlier work and apply the Bright-Ortiz-Zakharov continuum Beck theorem for hyperplanes together with Marstrand projection and slicing to push-forward measures coming from the volume map on (k+1)-tuples. They also flag a possible elementary route via the Fubini property for Hausdorff dimension from Héra-Keleti-Máthé. The abstract is direct about what is new and what external results are used. The extension itself looks like a natural incremental step that fills the gap between the boundary case and higher dimensions. The soft spot is the applicability of the hyperplane incidence result to this nonlinear volume functional. The level sets and incidence geometry for simplex volumes are not obviously the same as for linear hyperplanes, so the non-degeneracy or transversality conditions may require extra checking in the intermediate range. The abstract names the theorems but leaves the adaptation details to the body; until those are seen, the central bound rests on an unverified transfer. No circularity or invented entities appear in what is shown. This is for readers already working on Falconer-type problems and geometric measure theory in Euclidean space. Someone tracking dimensional thresholds for configurations will get direct value from the stated range extension. It deserves peer review because the core claim is a genuine step beyond the cited prior result and the tools are standard, even if referees will want the adaptation spelled out and checked against the stress-test concern.

Referee Report

1 major / 2 minor

Summary. The manuscript extends Shmerkin-Yavicoli's sharp threshold result for the Falconer-type simplex volume problem from the case d = k+1 to the intermediate range k+1 ≤ d ≤ 2k, claiming a dimensional threshold of k in this range and a non-trivial threshold of d-k when d > 2k. The argument applies the continuum Beck-type theorem for hyperplanes from Bright-Ortiz-Zakharov together with Marstrand projection and slicing theorems to push-forward measures induced by the (k+1)-volume map on (k+1)-tuples from E; an alternative elementary approach is also investigated under the Fubini property for Hausdorff dimension.

Significance. If the central claims are established, the work provides a meaningful interpolation between the sharp d = k+1 case and higher-dimensional regimes for a nonlinear higher-order configuration problem, strengthening the toolkit for Falconer-type questions involving simplex volumes. The explicit use of Bright-Ortiz-Zakharov and Marstrand results, if the transfer is justified, constitutes a technical strength; the Fubini-property investigation adds an independent perspective that could be useful even if the main argument requires adjustment.

major comments (1)

[Proof of Theorem 1.1 (or equivalent main result section)] The central extension to k+1 ≤ d ≤ 2k and the threshold d-k for d > 2k rests on transferring the Bright-Ortiz-Zakharov continuum Beck theorem (stated for hyperplane incidences) and Marstrand projection/slicing theorems to the nonlinear Vol_{k+1} functional and its induced measures. The manuscript must explicitly verify that the requisite non-degeneracy, curvature, and transversality conditions hold for this higher-order map in the intermediate-dimensional regime; without such a check the dimensional thresholds do not automatically carry over from the linear setting.

minor comments (2)

[Section discussing the Fubini property] Clarify the precise statement of the Fubini property for Hausdorff dimension as used in the elementary approach, including any dependence on the ambient dimension d.
[Introduction and preliminaries] Ensure all citations to Shmerkin-Yavicoli, Bright-Ortiz-Zakharov, and Marstrand include the exact theorem numbers or statements being invoked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive feedback. We appreciate the recognition of the work's contribution in extending the sharp threshold results to intermediate dimensions. We address the major comment below and will revise the manuscript to incorporate the suggested clarification.

read point-by-point responses

Referee: [Proof of Theorem 1.1 (or equivalent main result section)] The central extension to k+1 ≤ d ≤ 2k and the threshold d-k for d > 2k rests on transferring the Bright-Ortiz-Zakharov continuum Beck theorem (stated for hyperplane incidences) and Marstrand projection/slicing theorems to the nonlinear Vol_{k+1} functional and its induced measures. The manuscript must explicitly verify that the requisite non-degeneracy, curvature, and transversality conditions hold for this higher-order map in the intermediate-dimensional regime; without such a check the dimensional thresholds do not automatically carry over from the linear setting.

Authors: We agree that an explicit verification of the non-degeneracy, curvature, and transversality conditions is required to rigorously justify the transfer from the linear hyperplane setting of Bright-Ortiz-Zakharov to the nonlinear Vol_{k+1} map. In the current draft these properties are used implicitly via the smoothness of the volume functional away from lower-dimensional degeneracies and the dimension hypotheses on E, but we acknowledge that a dedicated check would strengthen the argument. In the revised manuscript we will insert a new lemma (in the section containing the proof of Theorem 1.1) that directly confirms the required conditions hold for the (k+1)-volume map when k+1 ≤ d ≤ 2k, drawing on the affine invariance of Vol_{k+1} and standard transversality estimates for generic projections. This addition will make the dimensional thresholds fully justified without altering the overall strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: extension relies on external theorems (Shmerkin-Yavicoli, Bright-Ortiz-Zakharov, Marstrand)

full rationale

The paper extends the Shmerkin-Yavicoli threshold for simplex volumes from the case d = k+1 to intermediate dimensions k+1 ≤ d ≤ 2k (and d-k for d > 2k) by invoking the continuum Beck-type theorem for hyperplanes from Bright, Ortiz and Zakharov together with Marstrand projection and slicing theorems applied to the push-forward measures from the (k+1)-volume map. These are independent external results with no reduction to quantities defined inside the present paper, no self-citations that are load-bearing, and no fitted parameters or ansatzes smuggled via prior work by the same authors. The derivation chain therefore remains non-circular and self-contained against the cited benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from geometric measure theory rather than introducing new free parameters or invented entities; the central claim depends on the transfer of projection and slicing theorems to the simplex-volume measure.

axioms (2)

standard math Marstrand's projection and slicing theorems apply to the Hausdorff dimension of the set of simplex volumes
Invoked in the abstract as a crucial part of the argument for the intermediate-dimension case.
domain assumption The continuum Beck-type theorem for hyperplanes from Bright, Ortiz and Zakharov transfers to the simplex-volume setting
Cited as a crucial part of the argument; if the transfer fails the extension does not hold.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We extend their result to k+1 ≤ d ≤ 2k and obtain a non-trivial dimensional threshold d-k when d>2k... application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems.

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

A Fubini-type theorem for Hausdorff dimension

H´ era, K., Keleti, T., M´ ath´ e, A. A Fubini-type theorem for Hausdorff dimension. JAMA 152, 471–506 (2024). https://doi.org/10.1007/s11854-023-0302-3

work page doi:10.1007/s11854-023-0302-3 2024
[2]

Bright, A

P. Bright, A. Ortiz, D. Zakharov, A Continuum Beck-type Theorem for Hyperplanes (2025), arXiv:2510.10907

work page arXiv 2025
[3]

X. Du, Y. Ou, K. Ren, R. Zhang, Weighted refined decoupling estimates and application to Falconer distance set problem (2023), arXiv:2309.04501

work page arXiv 2023
[4]

M. B. Erdo˘ gan, D. N. Hart, A. Iosevich, Multiparameter projection theorems with applications to sums- products and finite point configurations in the Euclidean setting, inRecent advances in harmonic analysis and applications, 93–103, Springer Proc. Math. Stat., 25, Springer, New York, 2013

work page 2013
[5]

Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1985

K. Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1985

work page 1985
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Falconer,Fractal geometry, Mathematical foundations and applications, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014

K. Falconer,Fractal geometry, Mathematical foundations and applications, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. 16

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Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika29(1982), no

K. Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika29(1982), no. 1, 109–115

work page 1982
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K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika32(1985), no. 2, 206–212

work page 1985
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B. Galo, A. McDonald, Volumes spanned byk-point configurations inR d, J. Geom. Anal.32(2022), no. 1, Paper No. 23, 26 pp

work page 2022
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Grafakos, A

L. Grafakos, A. Greenleaf, A. Iosevich, E. Palsson, Multilinear generalized Radon transforms and point configurations, Forum Math.27(2015), no. 4, 2323–2360

work page 2015
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Greenleaf, A

A. Greenleaf, A. Iosevich, M. Mourgoglou, On volumes determined by subsets of Euclidean space, Forum Math.27(2015), no. 1, 635–646

work page 2015
[12]

Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Onk-point configuration sets with nonempty interior, Mathematika 68(2022), no. 1, 163–190

work page 2022
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Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Nonempty interior of configuration sets via microlocal partition optimization, Math. Z.306(2024), no. 4, Paper No. 66, 20 pp

work page 2024
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Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Realizing trees of configurations in thin sets, Pacific J. Math.335 (2025), no. 2, 355–372

work page 2025
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L. Guth, A. Iosevich, Y. Ou, H. Wang, On Falconer’s distance set problem in the plane, Invent. Math.219 (2020), no. 3, 779–830

work page 2020
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London Math

J.M Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3)4(1954), 257–302

work page 1954
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J.M Marstrand, The dimension of Cartesian product sets, Math. Proc. of the Cambridge Philos. Soc.50 (1953), 198–202

work page 1953
[18]

Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ

P. Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ. Press, Cambridge, 2015

work page 2015
[19]

Mattila,Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ

P. Mattila,Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ. Press, Cambridge, 1995

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Mattila, Dimension, Projections, Intersections, and Besicovitch Sets

P. Mattila, Dimension, Projections, Intersections, and Besicovitch Sets. New Trends in Applied Harmonic Analysis,2(2019), 129–157

work page 2019
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McDonald, Areas spanned by point configurations in the plane, Proc

A. McDonald, Areas spanned by point configurations in the plane, Proc. Amer. Math. Soc.149(2021), no. 5, 2035–2049

work page 2021
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Shmerkin, A

P. Shmerkin, A. Yavicoli, On the volumes of simplices determined by a subset ofR d, Ann. Fenn. Math.50 (2025), no. 1, 97–108. (J. Gaitan)Department of Mathematics, Virginia Tech, Blacksburg, V A Email address:jogaitan@vt.edu (E. Palsson)Department of Mathematics, Virginia Tech, Blacksburg, V A Email address:palsson@vt.edu

work page 2025

[1] [1]

A Fubini-type theorem for Hausdorff dimension

H´ era, K., Keleti, T., M´ ath´ e, A. A Fubini-type theorem for Hausdorff dimension. JAMA 152, 471–506 (2024). https://doi.org/10.1007/s11854-023-0302-3

work page doi:10.1007/s11854-023-0302-3 2024

[2] [2]

Bright, A

P. Bright, A. Ortiz, D. Zakharov, A Continuum Beck-type Theorem for Hyperplanes (2025), arXiv:2510.10907

work page arXiv 2025

[3] [3]

X. Du, Y. Ou, K. Ren, R. Zhang, Weighted refined decoupling estimates and application to Falconer distance set problem (2023), arXiv:2309.04501

work page arXiv 2023

[4] [4]

M. B. Erdo˘ gan, D. N. Hart, A. Iosevich, Multiparameter projection theorems with applications to sums- products and finite point configurations in the Euclidean setting, inRecent advances in harmonic analysis and applications, 93–103, Springer Proc. Math. Stat., 25, Springer, New York, 2013

work page 2013

[5] [5]

Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1985

K. Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1985

work page 1985

[6] [6]

Falconer,Fractal geometry, Mathematical foundations and applications, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014

K. Falconer,Fractal geometry, Mathematical foundations and applications, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. 16

work page 2014

[7] [7]

Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika29(1982), no

K. Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika29(1982), no. 1, 109–115

work page 1982

[8] [8]

K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika32(1985), no. 2, 206–212

work page 1985

[9] [9]

B. Galo, A. McDonald, Volumes spanned byk-point configurations inR d, J. Geom. Anal.32(2022), no. 1, Paper No. 23, 26 pp

work page 2022

[10] [10]

Grafakos, A

L. Grafakos, A. Greenleaf, A. Iosevich, E. Palsson, Multilinear generalized Radon transforms and point configurations, Forum Math.27(2015), no. 4, 2323–2360

work page 2015

[11] [11]

Greenleaf, A

A. Greenleaf, A. Iosevich, M. Mourgoglou, On volumes determined by subsets of Euclidean space, Forum Math.27(2015), no. 1, 635–646

work page 2015

[12] [12]

Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Onk-point configuration sets with nonempty interior, Mathematika 68(2022), no. 1, 163–190

work page 2022

[13] [13]

Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Nonempty interior of configuration sets via microlocal partition optimization, Math. Z.306(2024), no. 4, Paper No. 66, 20 pp

work page 2024

[14] [14]

Greenleaf, A

A. Greenleaf, A. Iosevich, K. Taylor, Realizing trees of configurations in thin sets, Pacific J. Math.335 (2025), no. 2, 355–372

work page 2025

[15] [15]

L. Guth, A. Iosevich, Y. Ou, H. Wang, On Falconer’s distance set problem in the plane, Invent. Math.219 (2020), no. 3, 779–830

work page 2020

[16] [16]

London Math

J.M Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3)4(1954), 257–302

work page 1954

[17] [17]

J.M Marstrand, The dimension of Cartesian product sets, Math. Proc. of the Cambridge Philos. Soc.50 (1953), 198–202

work page 1953

[18] [18]

Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ

P. Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ. Press, Cambridge, 2015

work page 2015

[19] [19]

Mattila,Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ

P. Mattila,Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ. Press, Cambridge, 1995

work page 1995

[20] [20]

Mattila, Dimension, Projections, Intersections, and Besicovitch Sets

P. Mattila, Dimension, Projections, Intersections, and Besicovitch Sets. New Trends in Applied Harmonic Analysis,2(2019), 129–157

work page 2019

[21] [21]

McDonald, Areas spanned by point configurations in the plane, Proc

A. McDonald, Areas spanned by point configurations in the plane, Proc. Amer. Math. Soc.149(2021), no. 5, 2035–2049

work page 2021

[22] [22]

Shmerkin, A

P. Shmerkin, A. Yavicoli, On the volumes of simplices determined by a subset ofR d, Ann. Fenn. Math.50 (2025), no. 1, 97–108. (J. Gaitan)Department of Mathematics, Virginia Tech, Blacksburg, V A Email address:jogaitan@vt.edu (E. Palsson)Department of Mathematics, Virginia Tech, Blacksburg, V A Email address:palsson@vt.edu

work page 2025