We introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r) α balls of radius r. To each θ ∈ (0, 1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/ log r = θ. The resulting 'dimension spectrum' (as a function of θ) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding 'lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum.We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension and the provision of new bi-Lipschitz invariants.

In a previous paper we introduced a new 'dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers.2010 Mathematics Subject Classification. Primary: 28A80. Secondary: 37C45, 82B43.

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including the presence (or lack of) arithmetic progressions (or patches in dimensions at least 2), the structure of tangent sets, and the Assouad dimension.We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to 'asymptotically' containing arbitrarily large arithmetic patches.We present some applications of our results concerning sets of integers, which include a weak solution to the Erdös-Turán conjecture on arithmetic progressions.

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the 'upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the quasi-Assouad dimension at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu.We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.Mathematics Subject Classification 2010: primary: 28A80.

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Federated Learning (FL) bridges the gap between collaborative machine learning and preserving data privacy. To sustain the long-term operation of an FL ecosystem, it is important to attract high-quality data owners with appropriate incentive schemes. As an important building block of such incentive schemes, it is essential to fairly evaluate participants’ contribution to the performance of the final FL model without exposing their private data. Shapley Value (SV)-based techniques have been widely adopted to provide a fair evaluation of FL participant contributions. However, existing approaches incur significant computation costs, making them difficult to apply in practice. In this paper, we propose the Guided Truncation Gradient Shapley (GTG-Shapley) approach to address this challenge. It reconstructs FL models from gradient updates for SV calculation instead of repeatedly training with different combinations of FL participants. In addition, we design a guided Monte Carlo sampling approach combined with within-round and between-round truncation to further reduce the number of model reconstructions and evaluations required, through extensive experiments under diverse realistic data distribution settings. The results demonstrate that GTG-Shapley can closely approximate actual Shapley values, while significantly increasing computational efficiency compared to the state of the art, especially under non-i.i.d. settings.

Abstract. We provide estimates for the dimensions of sets in R which uniformly avoid finite arithmetic progressions. More precisely, we say F uniformly avoids arithmetic progressions of length k ≥ 3 if there is an ε > 0 such that one cannot find an arithmetic progression of length k and gap length ∆ > 0 inside the ε∆ neighbourhood of F . Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of k and ε. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension.We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a 'reverse Kakeya problem': we show that if the dimension of a set in R d is sufficiently large, then it closely approximates arithmetic progressions in every direction. Almost arithmetic progressions and dimensionArithmetic progressions are fundamental objects across mathematics and conditions which either force them to exist (or not exist) within a given set are of particular interest. For example, Szemerédi's seminal theorem [Sz] states that if A ⊂ N has positive upper density, then A contains arbitrarily long arithmetic progressions. We say a set {a i } k−1 i=0 ⊂ R is an arithmetic progression (AP) of length k if there exists ∆ > 0 such that a i = a 0 + i∆, for i = 1, 2, . . . , k − 1. We say ∆ is the gap length of the arithmetic progression. We are primarily interested in sets which uniformly avoid arithmetic progressions and for this reason it is useful to introduce a weaker notion of 'almost arithmetic progressions'. In particular, given ε ≥ 0 we say that {b i } k−1 i=0 ⊂ R is a (k, ε)-AP if there exists an arithmetic progression {a i } k−1 i=0 with gap length ∆ > 0 such thatfor all i = 0, 2, . . . , k − 1. Thus there is an arithmetic progression of length k and gap length ∆ > 0 inside the closed ε∆ neighbourhood of a (k, ε)-AP. We think of a set F ⊂ R as uniformly avoiding arithmetic progressions of length k if, for some ε > 0, it does not contain any (k, ε)-APs. Note that (k, 0)-APs are simply the usual arithmetic progressions of length k.

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can analogously be obtained as the minimal dimension of a microset. In particular, the maximum and minimum exist. We also show that for an arbitrary Fσ set ∆ ⊆ [0, d] containing its infimum and supremum there is a compact set in [0, 1] d for which the set of Hausdorff dimensions attained by its microsets is exactly equal to the set ∆. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.2010 Mathematics Subject Classification. Primary: 28A80, Secondary: 28A78.