A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

Mounting evidence suggests that Latin American democracies are characterized by politics and societies becoming more divisive, confrontational, and polarized. This process, which we define here as the “new polarization” in Latin America, seems to weaken the ability of democratic institutions to manage and resolve social and political conflicts. Although recent scholarship suggests that polarization is integral to contemporary patterns of democratic “backsliding” seen in much of the world, this new polarization in the region has not yet received systematic scholarly attention. Aiming to address this gap in the literature, the different contributions in this special issue revise the conceptualization, measurement, and theory of a multidimensional phenomenon such as polarization, including both its ideological and affective dimensions, as well as perspectives at the elite and mass levels of analysis. Findings shed light on the phenomenon of polarization as both a dependent and an independent variable, contributing to comparative literature on polarization and its relationship to democratic governance.

After tracing the history of the term “spirituality” and the discipline of spirituality up to the mid-twentieth century, this article describes the contemporary understanding of spirituality as lived religious experience and of the academic discipline which studies this subject. This phenomenology of the discipline grounds a position on the relationship between lived spirituality and theology on the one hand, and the academic disciplines of spirituality and theology on the other.

Father Ed Donelan came to New Mexico from Massachusetts. The priest worked as a chaplain at a facility for “juvenile delinquents,” and later ran a home for boys he called the Hacienda de los Muchachos. Donelan sexually abused youth at both facilities. This essay considers how Donelan leveraged New Mexico’s juvenile justice and habilitation systems, and racial inequities baked into them, to abuse young people. Within those systems, a Catholic culture of clericalism granted Donelan unlimited access to youth, and enabled him to move unchecked between spiritual and quasi-parental forms of authority. Donelan’s story shows that clericalism is not a one-size-fits-all problem; it manifests differently in relation to different communities. Here clericalism intersects with place-based power structures of race and colonialism to damage in locally specific ways. Donelan’s case demonstrates that scholars who study clerical sexual abuse need to pay attention not only to priests, but also to church and state institutions that rendered certain populations of children especially vulnerable to their bad actions.

In the context of big data, the rise of digital inclusive finance is driving the digital transformation of various industries, which in turn, has a profound impact on labour employment. Based on the panel data of 258 prefecture-level cities in China from 2011 to 2021, this article evaluates the impact of digital financial inclusion on the employment structure of the labour force and investigates its spatial spillover effects by constructing a spatial Durbin model and a mediation model. The study finds that: (1) Digital financial inclusion increases the employment demand for high-skilled and low-skilled labour and decreases the employment demand for middle-skilled labour, that is, the employment structure of the labour force shows a polarisation trend. (2) There is a positive spatial spillover effect of the impact of the development of digital financial inclusion on the demand for high- and low-skilled labour. (3) Financial digitisation plays an intermediary role in the process of digital financial inclusion affecting the employment structure of the labour force. Taking advantage of the dividends brought about by the development of digital inclusive finance and formulating policies according to the actual situation of each region indicate an undeniable choice for achieving full employment.