arXiv subjects

Samuel J. Ferguson

Publications and source records attributed to Samuel J. Ferguson.

8 recordsLinked to original sources

arXiv2026

Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements

We construct an explicit 64-point subset of $\mathbb F_2^9$ which tiles by translation but has no linear-subspace tiling complement. Together with our earlier dimension-eight theorem and a product argument, this shows that every nonempty tile in $\mathbb F_2^d$ has a linear-subspace complement precisely when $d\le8$. The example is spectral, so it is not a counterexample to Fuglede's conjecture. Its construction uses six three-dimensional subspaces of $\mathbb F_2^6$, specified by a small table. We classify all its tiling complements and spectra: there are exactly 512 complements and 64 spectra, with one translation class of each. Every spectrum has normalized Wiener norm $15/8$. An appendix proves that Goyal's related construction is affinely inequivalent but has the same tiling complements and spectra. The construction and both classifications are proved from first principles. We also give elementary proofs of the positive assertion in dimensions at most seven and for all tile sizes other than sixteen in dimension eight. The remaining case uses our previously established theorem based on the classification of perfect binary codes of length fifteen.

math.CO↗

arXiv2026

Fuglede's conjecture holds in eight dimensions over the field with two elements

Fuglede's conjecture in a finite abelian group says that a nonempty subset tiles by translation if and only if it is spectral, meaning that the space of complex-valued functions on the subset has an orthogonal basis consisting of restrictions of group characters. We prove the conjecture in the eight-dimensional vector space over the field with two elements. In fact, every spectral set and every tile in this space has a tiling complement which is a linear subspace. We first prove that the Fourier support of a nonempty proper subset of density different from one half contains a two-dimensional subspace. Consequently, every spectral set of density one quarter has a subspace tiling complement. This gives a simpler proof of Kadowaki's dimension-seven result, replacing his use of the classification of Hadamard matrices of order 32 by the classical five classes of order 16. Our dimension-eight proof uses exact finite verifications involving Hadamard matrices of order 32 and perfect codes of length 15. We include the proofs of the reductions used from Kadowaki's preprint, so that neither positive result depends on his dimension-seven theorem. Finally, we give an elementary spectral example of density one eighth in dimension nine whose Fourier support contains no three-dimensional subspace. Thus, the Fourier-support argument at quarter density does not extend in this form to eighth density. Dimension nine is the only case still unresolved over the field with two elements, and this obstruction shows why the present Fourier-support method cannot resolve it as it stands.

math.CO↗

arXiv2026

Fuglede's conjecture holds in $\mathbb{F}_{p}^{d}$ for $p=2,\,d=8$ and $p=11,\,d=3$

Fuglede's conjecture in $\mathbb{F}_p^d$ says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in $\mathbb{F}_2^8$ and $\mathbb{F}_{11}^3$. Over $\mathbb{F}_2$ we show more generally that the Fourier support of a nonconstant indicator of density different from one half contains a plane. Consequently every spectral set of density one quarter has a subspace tiling complement. This supplies one of the dimension-eight reductions left by Kadowaki; the others are completed using classified Hadamard matrices and perfect binary codes. In $\mathbb{F}_2^9$ we prove that every tile is spectral and that every spectral set has power-of-two cardinality. All spectral cardinalities except $64$ are shown to tile, and we give further reductions of the remaining case. We also construct a spectral set of density one eighth whose Fourier support contains no three-dimensional subspace. For $\mathbb{F}_{11}^3$ we extend the geometric method of Fallon, Mayeli, and Villano by using moment identities, polynomial plane profiles, weighted incidence counts, and exact finite computations. The computations and their verification programs are described in the appendices.

math.CO↗

arXiv2020

Obamacare and a Fix for the IRS Iteration

We model the quantities appearing in Internal Revenue Service (IRS) tax guidance for calculating the health insurance premium tax credit created by the Patient Protection and Affordable Care Act, also called Obamacare. We ask the question of whether there is a procedure, computable by hand, which can calculate the appropriate premium tax credit for any household with self-employment income. We motivate current IRS tax guidance, which has had self-employed taxpayers use a fixed point iteration to calculate their premium tax credits since 2014. Then, we give an example showing that the IRS iteration can lead to a divergent sequence of iterates. As a consequence, IRS guidance does not calculate appropriate premium tax credits for tax returns in certain income intervals, adversely affecting eligible beneficiaries. A bisection procedure for calculating premium tax credits is proposed. We prove that this procedure calculates appropriate premium tax credits for a model of simple tax returns. This is generalized to the case where premium tax credits are received in advance, which is the most common one in applications. We outline the problem of calculating appropriate premium tax credits for models of general tax returns. While the bisection procedure will work with the tax code in its current configuration, it could fail, eg, in states which have not expanded Medicaid, if a new deduction with certain properties were to arise.

q-fin.GN↗

arXiv2019

Higher-order linearization and regularity in nonlinear homogenization

We prove large-scale $C^\infty$ regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian $\bar{L}$, (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale $C^{0,1}$-type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations---with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.

math.AP↗

arXiv2018

A One-Sentence Line-of-Sight Proof of the Extreme Value Theorem

We give a one-sentence proof that a continuous real-valued function f on a closed, bounded interval attains a maximum value, by the following device. We define x in [a, b] to be a lookout point if f(t) does not exceed f(x) whenever t lies in [a, x). Letting c be the maximum of the set of lookout points, we prove that f(c) is the maximum value of f.

math.HO↗

arXiv2018

The Affordable Care Act and the IRS Iterative Fixed Point Procedure

We model the quantities appearing in Internal Revenue Service (IRS) tax guidance for calculating the health insurance premium tax credit created by the Patient Protection and Affordable Care Act, also called Obamacare. We ask the question of whether there is a procedure, computable by hand, which can calculate the appropriate premium tax credit for any household with self-employment income. We give an example showing that IRS tax guidance, which has had self-employed taxpayers use an iterative fixed point procedure to calculate their premium tax credits since 2014, can lead to a divergent sequence of iterates. As a consequence, IRS guidance does not calculate appropriate premium tax credits for tax returns in certain income intervals, adversely affecting eligible beneficiaries. A bisection procedure for calculating premium tax credits is proposed. We prove that this procedure calculates appropriate premium tax credits for a model of simple tax returns; and apparently, this procedure has already been used to prepare accepted tax returns. We outline the problem of finding a procedure which calculates appropriate premium tax credits for models of general tax returns. While the bisection procedure will work with the tax code in its current configuration, it could fail, in states which have not expanded Medicaid, if a certain deduction were to revert to an earlier form. Future policy objectives might also lead to further problems.

q-fin.GN↗

arXiv2018

A One-Sentence Inverse Image Proof of Lusin's Theorem

Standard proofs of Lusin's theorem, using simple functions, are sometimes quite elaborate. Here, we give a one-sentence proof of Lusin's theorem. We do not believe our approach, by way of inverse images, is new. However, this particular proof is strikingly clear and succinct.

math.CA↗