We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots \}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon \gt 0$,

\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*}

This answers a question of Erdős.

In the 1950s and 1960s, changes in the international situation, such as decolonization in Asia, led some Australians to question the usefulness of keeping the ‘White Australia Policy’, the basis for the country’s immigration system since Federation in 1901. Some argued that Australia’s international reputation, especially with newly independent countries in Asia and Africa, could be harmed by the maintenance of this policy. Events such as the Sharpeville Massacre in apartheid South Africa in 1960 caused further introspection into Australia’s racialized system. However, as pushes to reform the policy grew, others used international events to resist reforms. Using speeches by politicians and documents produced by policymakers, this article will show how events, such as the Notting Hill riots in Britain, the Little Rock controversy in the United States, and the Sharpeville massacre, were used as warnings about Australia potentially introducing similar ‘racial problems’ if it allowed more non-white migration. This article argues that these concerns tapped into a transnational whiteness that shared anxieties about decolonization, civil rights and non-white immigration in the post-war period, bringing a racialized solidarity forged at the turn of the twentieth century into the Cold War era.

The failure of the proposed Equal Rights Amendment (ERA) has been attributed to various organized interests, including the New Right and insurance companies. This study examines trends in lobby efforts regarding the amendment and correlations between lobby efforts and roll call votes among state legislators. Lobbyists active on the amendment appeared most often in states they perceived were most likely to approve. A second data set consisting of 6,952 votes reveals that explicitly pro- and anti-ERA lobby efforts were correlated with votes cast only by Republican state legislators. Lobby efforts by insurance companies were not correlated with any votes. The efforts of pro- and anti-amendment lobbyists, however, likely had no effect on the ultimate fate of the proposed amendment. Women and non-white legislators voted more often for the amendment, regardless of party. Moreover, changes in public support for the amendment led to partisan differences in legislators’ votes.

This paper explores the relationship between entrepreneurship, measured by the number of new firms per million inhabitants, and modern economic growth in Spain between 1886 and 2000. Following Audretsch and Keilbach’s methodology, our analysis seems to confirm that entrepreneurship has had a positive and statistically significant effect on GDP per capita and labor productivity. This finding challenges the traditional view that the entrepreneurial factor has hindered the country’s economic growth. Additionally, using data on the size and legal form of start-up firms, our results suggest that neither characteristic has been an important driver of Spain’s long-term economic growth. However, we find that the impact of both variables differs depending on the years studied. To our knowledge, this study is the first attempt to test econometrically the long-term contribution of entrepreneurship to Spain’s economic growth.

The quality of information that informs decisions in expert domains such as law enforcement and national security often requires assessment based on meta-informational attributes such as source reliability and information credibility. Across 2 experiments with intelligence analysts (n = 74) and nonexperts (n = 175), participants rated the accuracy, informativeness, trustworthiness, and usefulness of information varying in source reliability and information credibility. The latter 2 attributes were communicated using ratings from the Admiralty Code, an information-evaluation system widely used in the defence and security domain since the 1940s. Ratings of accuracy, informativeness, and likelihood of use were elicited as repeated measures to examine intraindividual reliability. Across experiments, intraindividual reliability was best when levels of source reliability and information credibility were moderately consistent compared to when they were maximally inconsistent (i.e., one low and one high) or maximally consistent (both high or low). As well, trustworthiness ratings depended more on source reliability than on information credibility. Finally, the likelihood of using information was consistently predicted by accuracy ratings and not by judged informativeness or trustworthiness. The current findings offer insights into the ability of experts and novices to reliably use information-evaluation systems for structuring human judgments about intelligence.

Research on citizens’ democratic support predominantly relies on surveys. However, the possibility of social desirability biases (SDBs) raises doubts about whether such instruments capture sincere attitudes. We search for evidence of SDB in measures of democratic attitudes in three studies. The first two leverage variations in survey mode (self-completion vs. face-to-face) in the European Social Survey’s Democracy module, drawing on evidence that interviewer absence encourages voicing socially undesirable opinions. The third uses a double-list experiment to estimate the prevalence of an anti-democratic attitude. Using data from as many as 24 European countries, we find no evidence that SDB inflates survey measures of democratic attitudes. These results contribute to our understanding of democratic attitudes and to methodological toolkit for interested scholars.