The Freiman--Ruzsa Theorem over Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion

The Perspective of People Receiving Food Support

The research examines the perspective of Israelis assisted by Non-governmental organization (NGOs) to promote their food security. This qualitative study employed the Interpretive Phenomenological Approach. The sample consisted of 16 recipients of food support aged 33-62, and they were interviewed in-depth semistructured protocol. Five main themes emerged: The background of food insecurity; Forms of food support; The experience of receiving food support; The effect of the assistance on food security; and the responsibility for food security. The participants indicated that the state welfare system should be responsible for food security, and they prefer payment cards that enable them to purchase food suitable to their individual needs while maintaining their dignity. Although the support provided does not entirely extricate families from food insecurity, they gain confidence by being able to receive food in times of hardship. The findings indicate the importance of considering the opinions of those in need when making policy decisions regarding food security

Patterns in Random Permutations

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests